



V 

































































































































* 

































. 

' 









































» 



NEW GENERAL THEORY OF 
MULTISTAGE AXIAL FLOW TURBOMACHINES 


BY 

DR.-ING. WALTER TRAUPEL 

U 

TRANSLATED BY DR. C. W. SMITH 
OP GENERAL ELECTRIC CORPORATION 



NAVSHIPS 250-445-1 
NAVY DEPARTMENT 
WASHINGTON 25, D.C. 


> i 

> 9 

> ■> 



2321 


The Bureau is indebted to Dr. C. W. 3nith of 
the General Electric Corporation for his excellent 
translation of this article. Treating as it does 
the design of compressors and turbines under a 
single theory, it should prove of interest and value 
to American designers. 


HEW GENERAL THEORY OF 


MULTISTAGE AXLAL FLOW TURBOMACHINES 


By 

Br.-ing. Walter Traupel 

U 


Thesis - Eidgen. Techn. Hochschole, Zurich 


Copyright, 19^2, hy AG. Gehr. Leemarm and Co. 

Published by AG. Gebr. Leemann and Company 
Zurich and Leipzig 


\ 


i 



To Johann Sebastian Bach 




* 

ii 


2321 



List of References Cited in Text 


J. Ackeret: Uber die Luftkrafte bei sehr grossen 
Geschwindigkeiten, insbesondere bei ebenen 
Stromungen (Concerning Air Forces at Very 
High Velocities, Particularly with Plane 
Flow). Helvetica physika Acta, Vol. I, 
page 301. Basel, 1928 

E. Amstutz: Zur theoretischen Vorausberechnung 
der Charakteristiken von spezifisch 
schnellaufenden Axialradem (On the 
Theoretical Prediction of the Characteristics 
of Specific High Speed Axial Flow Rotors). 
Stodola-Festschrift (Stodola Anniversary 
Volume). Zurich, 1929 

G. Darieus: Contribution au trace des aubes 
radiales de turbines (Contribution to the 
Design of Radial Turbine Blades). Stodola- 
Festschrift (Stodola Anniversary Volume). 

Zurich, 1929 

A. Egli: A Rational Representation of the Flow 
Performance of Reaction Steam-Turbine 
Blading. Journal of Applied Mechanics, 

Vol. 7, Ho. 1. New York, 1940 

C. Keller: Axialgeblase vom Standpuhkt der 

Tragflugeltheorie (Axial Compressors from 
the Standpoint of the Airfoil Theory). 

Thesis, Eidgen. Techn. Hochschule. Zurich, 

1934 

P. Lorrain: Les turbines a vapeur et a combustion 

interne (Steam and Internal Combustion Turbines). 
Paris, 1935 

N. W. Me Lachlan: Bessel Functions for Engineers. 
Oxford, 1934 

L. M. Milne-Thomson: Theoretical Hydrodynamics. 
London, 1938 

The National Advisory Committee for Aeronautics 
(NACA): Report No. 460. Washington, 1933 

E. Jahnke, F. Emde: Funktionentafeln (Tables of 
Functions). Leipzig, 1938 

E. Weinel: Beitrage zur rationellen Hydrodynamlk 
der Gitterstromung (Contributions to the 
Rational Hydrodynamics of Grid Flow). 

Ing. - Archiv., Vol. V, Page 91> Berlin, 1934 


iii 


2321 


CONTENTS 


Page 

Preface 

I. Introduction. 

1.1 Object and Nature of the 1 

New Theory 

1.2 Arrangement of the Book 1 

1.3 Notation 2 

1.4 Definitions 7 

II. Elementary Theory of Frictionless 
Flow Through the Stages of 
Turbomachines 

2.1 Notation and Assumptions 11 

2.2 Energy Transfer in a Stage 12 

Element 

2.3 Number and Choice of the 13 

Independent Characteristics 

of a Stage Element 

2.4 Determination of the 17 

Dependent Characteristics 

of a Stage Element 

2.5 General View of the Normal 20 

Stage Element 

2.6 General Considerations 21 

Pertaining to the Stage of 

Finite Radial Length 

2.7 The Normal Stage 24 

HI. Theory of Grid Testing and Loss 
Coefficients 

3.1 Notation and Assumptions 26 

3.2 Measurements Required to 27 

Determine the Grid 
Characteristics for an 
Incompressible Fluid 


iv 


Contents 

- 2 - 

Page 

3.3 

Discussion of Compressible 

Fluid Case 

35 

3.4 

Deflection Coefficient 

39 

3.5 

Grid Energy Loss Coefficient 

40 

3.6 

Grid Force Coefficients and 
the Loss Coefficients 

Derived from them 

4l 

3.7 

Relation Between Loss Coef¬ 
ficients 

43 

3.8 

Numerical Example. Data 
from tests made with an 
Accelerating Grid 

45 

17. Calculation of the Stage on the 

Basis of Grid Tests 


4.1 

Notation 

49 

4.2 

General Discussion of the 
Problem 

50 

4.3 

Validity of the Equations 
of Chapter II 

55 

4.4 

Matching of Two Grids in a 

Stage Element 

56 

/ 

4.5 

Efficiency of the Stage 

Element as a Function of 
the Grid Energy Loss 

Coefficients 

58 

4.6 

Efficiency of the Stage 

Element as a Function of 
the Drag-Lift (or Glide) 

Ratio 

6l 

4.7 

Efficiency of the Stage 

Element as a Function of 
the Friction Coefficients 

63 

4.8 

Basic Stage Efficiency 

63 

4.9 

Clearance Losses 

66 

4.10 

Wall Losses 

69 


* 


V 


2321 


Contents 


- 3 - 


Page 


4. 11 

Stage Efficiency, Pressure 
Coefficient, and Work Coef¬ 
ficient of the Stage 

72 

4.12 

Effective Change of Enthalpy. 
Thermodynamic Efficiency 

73 

Calculation of the Multistage 

Turbomachine- 


5.1 

Notation 

75 

5.2 

Thermodynamic Basis 


5.3 

The Energy Equations 

78 

5.4 

The Continuity Equation 

82 

5.5 

Calculation Procedure 

85 

5.6 

Numerical Example: 

Calculation of a Gas Turbine 

Unit 

88 

5.7 

Use of Dimensionless 

Variables in Multistage 

Blading Analysis 

92 

Theory of Three-Dimensional Flow of 
Non-Viscous Fluids Through Turbo¬ 
machine Stages — 


6.1 

Notation and Preliminary 

Remarks 

94 

6.2 

General Survey of the Phenomena 

95 

6.3 

General Discussion of the 

Problem 

9 6 

6.4 

Three-Dimensional Potential 

Flow with Inclined Stationary 
Blades 

103 

6.5 

Numerical Example for Section 

6.4 

110 

6.6 

Rotational Flow through the 
Homogeneous Stage 

112 

6.7 

Numerical Example for Section 

6.6 

119 

6.8 

Effect of Compressibility 

120 


vi 



2321 



HJHSTRA.TIQMS 




Page 

Fig. 1 

Meridian Passage and Meridian Streamline 
Pattern 

122 

Fig. 2 

Change of State in a Stage Element Shown 
on the Enthalpy-Entropy Chart 

123 

Fig. 3 

Velocity Diagrams (General Case) 

124 

Fig. 4 

Velocity Diagrams for the normal 

Stage Element 

125 

Fig. 5 

General Survey of Normal Stage Elements 

126 

Fig. 6 

Blade Grid 

127 

Fig. 7 

Blade Grid Used in Numerical Example 

127 

Fig. 8 

Velocity Diagram of Blade Grid 

128 

Fig. 9 

Developed Stage Element 

128 

Fig. 10 

Velocity Distribution Function Used 
in Numerical Example 

129 

Fig. 11 

Velocity Diagram of the General 

Cylindrical. Stage Element 

129 

Fig. 12 

Graphical Determination of the 

Associated Values of oC_and & 

oo h^oo 

130 

Fig. 13 

Diagram Used in Derivation of 

Clearance Loss Formula 

130 

Fig. 14 

Diagram Used in Derivation of 

Clearance Loss Formula 

131 

Fig. 15 

Extrapolation to Zero Clearance 

131 

Fig. 1 6 

Diagram Used in Derivation of 

Wall Loss Formula 

132 

Fig. 17 

Change of State in a Multistage Turbo¬ 
machine. Shown on the Enthalpy-Entropy 

Chart 

133 

Fig. 18 

Expansion in a Multistage Turbine Shown 
on the Temperature Entropy Chart 

134 

Fig. 19 

Multistage Blading. Division into 

Groups of Stages 

135 

Fig. 20 

* 

Velocity Diagrams (at Principal Diameter) 
for the Illustrative Numerical Example 

136 


vii 


2321 



Fig. 21 

Fig. 22 

Fig. 23 
Fig. 24 
Fig. 25 
Fig. 2 6 
Fig. 27 
Fig. 28 
Fig. 29 

Fig. 30 
Fig. 31 

Fig. 32 
Fig. 33 
Fig. 34 
Fig. 35 

Table 1 

Table 2 
Table 3 

Table 4 


Single Stationary Blade Grid in an 
Infinitely Long Meridian Passage with. 
Cylindrical Bounding Walls 

Coordinate System, Direction of the 
Velocity and Dotation Vectors, and 
Direction of the Gradient of the 
Total Pressure in the Clearance 
Space between the Rows of Blades 

Flow of an Incompressible Fluid 
Through a Group of Identical Stages 

Transition to the Limiting Case of 
an Infinitely Small Blade Pitch 

Stationary Grid with Curved, 

Inclined Blades 

Homogeneous Group of Stages with 
Inclined Stator Blades 

Compressor Blading with Inclined Stator 
Blades (numerical Example 6.5) 

Theory of Turbulent Flow, Coordinate 
System and Reference Planes 

Meridian Streamline Pattern, Variation 
of C a ^, C a g, y, and ^2 ^ ^ 

Reaction Turbine with Identical Unwarned 
Blades 

Integral F as a Function of B]_ 

Velocity Diagrams at Different Radii for 
Reaction Turbine with Identical Uhwarped 
Blades 

Variation of Degree of Reaction along the Blade 
Wind Tunnel with Grid Model 
Experimental Blade Grid 
Grid Test Results 


APPENDIX 


Relation between Grid Energy Loss 
Coefficient and Friction Coefficient 

Functions of Y 

Values of Function Used in 
Obtaining Number of Stages 

Heat Recovery and Heating Loss Factor 


viii 


rage 

137 

138 

138 

139 

140 

140 

141 

141 

142 

143 

144 

145 

146 

146 

147 

148 

149 

150 

151 


2321 



PREFACE 


This "book, which establishes the theory 
of multistage social flow turbomachines on a 
modern basis, grew out of studies made by the 
author in connection with his activities in 
the Educational Division of Gebruder Sulzer 
A.-G., Winterthur. The introductory chapter 

4* 

explains the objective of the new theory. 

The reader should note the following. 

The division into chapters and sections follows 
a decimal system. In each section the numbering 
of the equations begins anew with (l). Wherever 
in the text a reference such as "see Equation (5)” 
is made, the reference is to an equation in the 
same section. References to equations in other 
sections are in the form "Equation 4.3 (6)", meaning: 
Section 4.3, Equation (6). In the tables of 
notation there will occasionally be found 
references such as "(Def. 4.8)", meaning that 
the definition of the quantity in question will 
be found in the section given. 

I must not fail to thank Professor H. Quiby 
for the interest with which he has followed the 
work since its inception, and especially 
Gebruder Sulzer A.-G. for its kindness in making 
possible the publication of the work. 

W. Traupel 

Winterthur 
January, 1942 


ix 


2321 

























































I. HTTRODUCTION 


1.1 Object and Nature of 
The New Theory 


The objective of the theory here developed is to establish a method 
of design calculation, based upon modem concepts of fluid mechanics, which 
will be applicable to both axial flow turbines and axial flow compressors. 
Only a certain class of turbines is included; namely, those of which the 
individual stages, at least when taken in groups, have a certain similarity. 
(This similarity will be further discussed under the definition of homo¬ 
geneous groups of stages.) The theory does not include, therefore, turbines 
with partial arc of admission, Curtiss wheels, and these low pressure stages 
of steam turbines which operate with wet steam and have such changes of 
specific volume that every stage operates under entirely different conditions. 
The theory applies particularly to the gas turbine which, used in conjunction 
with the axial flow compressor, is rapidly assuming great importance. 

At present the calculation of multistage axial turbines is based 
upon customary steam turbine theory (which has several variations and in¬ 
cludes the results of a great number of theoretical, investigations on special 
problems), while a complete theory of the multistage axial compressor has not 
yet been published. The customary steam turbine theory is not consistent in 
many respects with the principles of modem fluid dynamics. It is, of course, 
conceivable that the data from modem aerodynamic research might be used to¬ 
gether with the methods of calculation hitherto employed, since experimental¬ 
ly determined coefficients must always be intro diced into the calculations. 
This would require merely the substitution of data from the later researches 
for the experimental coefficients which now occur in the equations. This 
procedure, however, is fundamentally unsatisfactory for the following reasons. 

First, the complication of the flow process is so great that in the 
case of machines with new types of blades it is impossible to predetermine 
the character of the flow, and hence it is impossible to predict the ef¬ 
ficiency with any degree of certainty. A very precise calculation of this 
kind can be made only if a sufficiently comprehensive series of tests with 
a given type of blade is available. However, since the usefulness of a 
theory is greatest for new developments which are only partially within the 
range of previous experience, the numerical accuracy of a calculation which 
is possible when the theory is used with adequate data is not the only cri¬ 
terion of its value. Of at least equal importance is its basic theoretical 
soundness. 

Secondly, it should be emphasized that the multistage axial flow 
compressor can theoretically be treated in exactly the same way as the 
turbine, since in operation and construction it is analogous to the multi¬ 
stage axial turbine. For this common treatment, however, the classical 
steam turbine theory is not suitable. Indeed, even in its own field this 
theory in its present state is strictly applicable only to impulse turbines 
and to noncondensing turbines with 50$ reaction. 

The object of this work, therefore, is to develop a theory which is 
theoretically exact so far as possible, and which at the same time is ap¬ 
plicable to both types of machines. 

1.2 Arrangement of the Book 

In Chapter 2 an elementary treatment of frictionless flow through 
the stages of turbo machines is given. The general method of treatment and 


1 


2321 


the characteristic quantities which are introduced in this chapter consti¬ 
tute the foundation of the theory and allow a general survey of all possi¬ 
ble types of homogeneous axial stages. The transition to flow with friction 
offers no further difficulty. 

Chapter 3 treats of the theory of grid (sometimes called lattice, 
or cascade) testing and considers the physical meaning of the grid coef¬ 
ficients which are later introduced into the calculation of the stage. An 
example of a grid test is given in connection with this treatment. For 
the further understanding of the theory it is not absolutely necessary to 
read the whole of Chapter 3. It is sufficient to know the meaning of the 
grid coefficients which are there introduced. 

Chapter h treats of the calculation of stage efficiency on the 
basis of grid measurements. 

Chapter 5 gives the process of calculation for determining dimensions 
and efficiency of the entire multistage machine on the basis of the stage 
efficiency. The methods given can be understood and used without the ne¬ 
cessity of reading Chapters 3 and 4, provided that the stage efficiency is 
given. 


Chapter 6 presents a somewhat more complete theory of the flow of 
a non-viscous fluid through a stage of a turbamachine. This theory does 
not make the usual assumption that the flow in the stage is along specified 
surfaces of revolution (mostly either cylindrical or conical), but de¬ 
termines the actual surfaces more definitely by calculation, treating both 
the case of potential flow and the case of rotational flow. Numerical 
examples are given and conclusions drawn about the effects of warped and 
unwarped blades, as well as of the angles of the blades. The previous 
chapters form by themselves a complete and usable theory without Chapter 6. 
However, Chapter 6 introduces a refinement in the theoretical treatment 
which can easily be carried over into all the previous analysis. 

1.3 Notation 

The following symbols are used throughout the entire book: 


A 


Chits 


m 

*8 

kcal 

sec 

min 

°C 


meter 

kilogram 

ki logram-calorie 

second 

minute 

degrees Kelvin 
(absolute centigrade 
temperature) 

Mechanical equivalent of heat = 

TfPf p ca7 


2 


2321 





% = 
G 

Had 


- D n±&s 


^subscript 


P (*) 


Inner (root or hub) diameter 
of meridian flow passage; 
i.e., of section taken in plane 
passing through axis of rotation 

N 

Outer (tip) diameter of 
meridian flow passage [m] 

Mean diameter [mj 

Mass flow ^kg sec~^*J 

Total adiabatic (isentropic) 
change of enthalpy [kcal kg" 3 ? 

Total actual change of enthalpy 
jjccal kg“^j 

Total effective energy input or 
output corresponding to the change 
of enthalpy |jccal kg"^j 

Velocity coefficient, the sub¬ 
script denoting the particular 
velocity in question 

Mechanical, work £kg m kg - -*- = mj 
Pressure energy per unit mass 
Qn 2 sec ~^J 


r (Translator's note: Although the quantity P, which is equal 
to pressure divided by mass density, is often used in American 
practice, there is no commonly accepted word for it, and its 
definition has therefore been literally translated. In the 
text it is usually called "pressure energy". It might also 
have been translated "pressure head", but it seems best to 
use this term only for pressure divided by specific weight. 

The dimensions of the quantity are therefore (length/time)2 # 
or velocity squared. Correspondingly, the "kinetic energy" 
in the text has the dimensions of velocity squared also). 


3 



Q 

R 

T 

* = D s/Dn 
c 


°p> C Y 


h(x) a A 3^_, 

2 g 


ad'* 1 ad 


lie 


Volume flow £m5 sec~lj 

Gas constant Jjcg m kg” 1 = 

Absolute temperature £°kJ 
Diameter ratio 

Absolute stream velocity sec -1 j 

Specific heat at constant pressure 
and constant volume, respectively 
|kcal kg“^ 

Acceleration of gravity = 9.81 
|ja sec-^j 

where x stands for any velocity 
|^kcal kg~^j 

Adiabatic (isentropic) change of 
enthalpy in stage, stationary blades, 
and moving blades, respectively 
(cf Fig. 2) — jjccal kg _1 J 

Actual change of enthalpy in 
stage (cf Fig. 2) 

£kcal kg -1 j 

change of enthalpy corresponding to 
energy conversion in stage (cf 
Fig. 2) jjccal fcg -1 j 

Blade radial height 

Speed of rotation jjnin*' 1 ] 


h 


2321 



Pressure 



Radius (measured from axis of 

rotation) £mj 

Blade chord \jz] 

(Note: this symbol has the 
meaning of "entropy" when 
used in the entropy diagram) 

Time [[sec] 

Blade pitch [[mj 

Number of stages 



Circulation 


Dynamic degree of reaction 
Total annular area of flow 
passages; i.e., annular area 
included between the circumfer¬ 
ential bounding surfaces of the 
passages in a plane perpendicular 
to the axis of rotation 



Specific weight 


Drag-lift ratio (reciprocal of 
lift-drag ratio) of the grid 
profile (sometimes called 
glide ratio) 

Lift and drag coefficients 
Tangential and normal force co¬ 
efficients (Def. 3.6 (6) (7)) 
Grid energy loss coefficient 
(Def« 3.5 (D) 

Basic loss coefficient of stage 


■S. 

^ap 

7o = 1 " ^0 

?. = 1 - ?e 

"7s 

r* 

7u 


7 


D 


k = c p/ c v 
* 

/* 

P 

r 


Loss coefficient of stage element 
Clearance loss coefficient 

Wall loss coefficient 

Basic efficiency of stage (Def. 

4.8) 

Efficiency of stage element 
(Def. 4.5 (2) (5)) 

Stage efficiency 

Thermodynamic efficiency (Def. 

4.12 (5) (4)) 

Blading efficiency (or 
"indicated" efficiency which, 
however, does not include effect 
of losses from leakage through 
shaft packing and from provisions 
for balancing thrust) 

Efficiency of diffuser following 
blading 

Friction coefficient (Def. 5.6 (11)) 
Ratio of specific heats 
Work coefficient (Def. 1.4 (4)) 
Density £kg m _J+ sec 2 J 

Kinematic degree of reaction 
(Def. 2.4 (1)) 

Whirl coefficient (or tangential 
velocity ratio) (Def. 1.4 (3)) 

Flow coefficient (or axial 
velocity ratio) (Def. 1.4 (2)) 


6 


2321 



Pressure coefficient (Def. 1.4 (5)) 
Functions of Y (Compare Table 2) 

Angular velocity / sec" 1 


Subscripts: 

T - Turbine 
K - Compressor 

H - Referred to principal stream surface 
S - Entire stage 
n - At inner diameter, or hub 
m - At mean diameter 
s - At outer diameter, or tip 
Special Symbols: 


Symbol * is used to denote the stationary blade 
grid, or stator 

Symbol " is used to denote the moving blade 
grid, or rotor 

Vector quantities are underlined. 


1.4 Definitions 


(a) Total Mechanical Energy Content of a Fluid Particle. By this 
is to be understood^the sum of the kinetic energy and the pressure energy 
given by P s J dp^ A potential energy content arising from a 

field of force is not included, although the actual process might be re¬ 
placed or represented by processes involving fields of force. It follows 
from this definition that the "total energy content" of the particle would 
be changed by passing through a field of force, as contrasted with the 
definition commonly used in hydrodynamics, whereby through the introduction 
of a potential energy a total energy content of a particle is so defined 
that it cannot be changed except by processes which involve losses. 
Thermodynamically, P is essentially equivalent to the enthalpy i. 


(b) Velocities, Stream Angles, Entropy. When velocities, stream 
angles, or entropy are spoken of as preceding the entrance or following the 
exit of a row of blades or blade grid, then, unless otherwise specified, it 
will always be understood that the quantity in question is to be the mean 
value taken along the blade pitch. As to how far from the blade annulus or 
grid these mean values are to be considered valid, compare Chapter 3. 


7 


2321 





(c) Rotor Velocity . If the. angular velocity of the rotor is given, 
then the tangential velocity at any point at a distance r from the axis 
of rotation can he easily calculated if the point is fixed in the rotor. 
This is called the rotor velocity at the point. (This is not to he confused 
with the tangential component of the stream velocity at that point.) 

(d) Velocity Ratio, Axial Velocity Ratio, Tangential Velocity 
Ratio . The velocity ratio at a point is the ratio of the fluid velocity, 
or of one of its components, to the rotor velocity at that point. In 
particular, the axial velocity ratio lj) is the ratio based on the axial 

component of the stream velocity, and the tangential velocity ratio T' is 
the ratio based on the tangential component. The axial velocity ratio is 
also known as the flow coefficient, and the tangential velocity ratio as 
the whirl coefficient. Aside from y? and T , the velocity ratios 

are expressed as K with a subscript, the subscript denoting the exact 
velocity in question. Hence 

K. = c/u, f z c a /u, X = c u /u (1)(2)(3) 

(e) Characteristic Velocity . Characteristic velocity is defined as 
the ratio of a velocity or a velocity component to any fixed reference 
velocity taken so that the ratio has a general application. 

(f) Stage Element . A stage element is an infinitesimal part of a 
stage which lies between two infinitely close stream surfaces (taken as 
surfaces of revolution) and is bounded by them (compare Fig. 1). A 
normal stage element is one in which the stream surfaces are cylindrical 
and in which the stage element is part of a homogeneous stage (see under i). 

(g) Stage Section . A stage section is an infinitesimal part of a 
stage which lies between two infinitely close co-axial circular cylinders 
(which need not necessarily be stream surfaces) and is bounded by them 
(compare Fig. 1). 

(h) Principal Stream Surfaces. One of the stream surfaces (taken 
as a surface of revolution) is selected as a surface of reference for the 
characteristic data of the stage. This is the principal stream surface 
of the stage. The surfaces chiefly used as principal stream surfaces are 
the Inner surface (root or hub), outer (housing), or a mean surface between 
these. 


(i) Homogeneous Stages. A homogeneous stage has the following 
property. In general the stream velocity in every plane perpendicular 
to the axis of rotation which it is possible to erect in the stage is a 
function of the radius r or a function of the radius ratio r/rg, where 

rg is the radius to the principal stream surface. Not only is the magni¬ 
tude of the velocity a function of r/rg, but also the angle which it makes 

with the plane and with the tangent of the circle passing through the point 
in question, so that its "direction" is also a function of r/r^. The 

direction is the same function of r/rg, both preceding the entrance to the 

stationary blades and following the discharge from the moving blades (and 
therefore also preceding the entrance to the next stationary blades). More¬ 
over, the magnitude is the same function of r/rg, except for a constant 

factor, and the velocities at any two points on the principal stream surface 
are proportional to the respective radii r~ . If these conditions are 


8 


2321 










exactly fulfilled, the stage is homogeneous. If they are exactly fulfilled 
only on the principal, stream surface and elsewhere only approximately, then 
the stage is only approximately homogeneous. With these assumptions the 
effect of the finite number of blades is made negligible. What the as¬ 
sumption of homogeneity really means is that the ratios and relations exist¬ 
ing at the entrance and exit of the Btage are the same. 

(k) Homogeneous Groups of Stages . By this will be understood a group 
of successive homogeneous stages with the self-evident additional stipulation 
that the ratios and relations existing at entrance to and exit from the 
stationary blades shall be the same. 

(l) Meridian Passage, Meridian Streamlines . If a longitudinal cross 
section (the plane passing through the axis of rotation) is taken through a 
multistage turbomachine, the flow path defined in this section by the outer 
and inner boundaries of the stationary and moving blade grids, respectively, 
is called the meridian passage (compare Fig. 1). The intersections of the 
stream surfaces with such a longitudinal (or meridian) plane (that is, the 
streamlines appearing in the meridian passage) are called the meridian stream¬ 
lines. 


(m) Dynamic Degree of Reaction. The dynamic degree of reaction of a 
stage element or stage section is the ratio of the pressure energy conversion 
in the rotor to the total pressure energy conversion in the stage. The 
kinematic degree of reaction will be defined later (see under 2.4). 


(n) Work Coefficient and Pressure Coefficient . Let ug be the rotor 
velocity (i.e., the-linear velocity of the rotor at that point) of a stage 
element or stage section at exit from a moving blade, let h^ be the 

adiabatic (isentropic) change of enthalpy in the stage, and let h e be the 

change of enthalpy corresponding to the effective work (compare Fig. 2). 

Then the following definitions may be written: 


Work Coefficient: /\. = ^6 



h(u£) 


00 


v 


Pressure Coefficient: 


Y 




^ad 


2 t 


h(u£) 


^P 


S- u 


The values of A and V at the principal stream surface may be con¬ 


sidered as characteristics of the stage. 


( 5 ) 


(o) Velocity Ratio. It is often more desirable, especially with 
turbines of 50$ reaction, to use instead of the pressure coefficient \f/ the 

velocity ratio JS corresponding to it. This is defined as follows. 



9 


2321 















It is to "be noted that VAP in the case of 50 $ reaction is that 

velocity with which the fluid would he discharged from the stationary 
blade grid with frictionless flow and zero initial velocity. The velocity 
ratio V will always be referred to the principal stream surface. It 

can be used as a "dimensionless tangential velocity" if it is necessary 
to deal with other processes carried out with the same blading voider 
different conditions (compare A. Egli: A Rational Representation of the 
Flow Performance of Reaction Steam-Turbine Blading). 


UO 


3321 


II. ELEMENTARY THEORY OF FRICTIONIESS FLOW 
THROUGH THE STAGES OF TURBO-MACHINE 


2.1 Notation. As sump t lone 


In addition to the symbols given under 1.3, which hold throughout 
the book, the following special symbols are used in this chapter. 

c 0 f , eg Absolute velocity at the entrance to a stationary blade 

grid, at the exit of a stationary blade grid, at the 
exit of a moving blade grid J" -lj 


c u0> c a0> °r0> e ^ c * 


Tangential, axial and radial components of co ; 
analogously for C]_ and eg f m sec" 1 ] 


o , 

00 


m lf mg 

u 


Mean ideal fluid velocity in stationary blade grid 
(absolute), and in moving blade grid (relative), 
respectively £ m sec" 1 ] 


Radius ratio defined by 2.2 ( 3 ) 

Tangential velocity [m sec -1 


w l^ Vg 


Relative entrance and discharge velocities of fluid 
in moving blade grid r m Bec -l] 


= Wu; K Woo = Vpo /u 


E r 

II 

<ri 

H 

1 

^2 Circulation coefficient 

y = r / r n 

Variable radius ratio 

II 

O 

Maximum radius ratio 

^ Of 

CM 

H 

* 

Angle of c Q , c-^ , eg ; as shown in Figure 3 . 


ft 2 


Angle « f w ( , w 2 ; as shown in Figure 3 . 

Angle of c^ w p9 ; as shown in Figure 4. 


Subscripts: 

0 - preceding the entrance to the stationary blade grid 
(Plane 0 in Figure 1) 

1 

1 - following the exit from the stationary blade grid 
(Plane 1 in Figure 1) 


11 



2 - following the exit from the moving "blade grid 
(Plane 2 in Figure 1) 

The asterisk (*) indicates "of the substituted cylindrical 
stage element" 

Caution: Xz , like all tangential components, is 

negative when the directions of the tangential component of the 

absolute fluid velocity and of the rotor velocity are opposite . 

In this chapter the following assumptions are made: 

(a) The fluid is considered to have zero viscosity. 

(b) The effect of the finite number of blades is neglected. The 
stream surfaces are assumed to be surfaces of revolution. 

(c) The stream surfaces - that is, the meridian streamline pattern - 
must be given. Whether the surfaces are b imply assumed, in the absence of 
more accurate data (for example, as cylindrical or conical), or whether 
they are known to some degree of approximation (as, for example, on the 
basis of the method outlined in Chapter Yl), is immaterial so far as this 
chapter is concerned. 

(d) If the stage is homogeneous, the total energy conversion is 
the same along every stream surface. The total work input or output in the 
stage is, therefore, the same per unit mass for every portion of the fluid. 

2.2 Energy Transfer in a Stage Element 

The energy transfer in a stage element is given by the Euler moment 
of momentum equation, which is expressed in terras of tangential velocities. 
The work done, expressed in heat units is 


he - A (u x c^ - U£ c^ ) „ (1) 

8 

whereas the total change of pressure energy (or head) is 

Haa =* (“l °ul - “s °u2 + 0 2 - °0 2 \ (2) 

8 2 

The dimensionless tangential and axial velocity ratios (or whirl and flow 
coefficients, definition 1.4d) are now introduced, as well as the following 
radius ratios 

”° : w ni ■ -g- (3) 

rQ , r-^ , r 2 are the radii of the stage element at the points 0, 1, 2. 

Using the definition equations of the velocity ratios, of the work coef¬ 
ficient \ and of the pressure coefficient T// from the equations of 1.4 

(1 to 5 inclusive), equations (1) and (2) may be written in dimensionless form. 


12 


2321 








2 K 2r l -*2> 


(*) 


A 


= 2 ( mi 2 z 1 . r 2 ) ( f 2 2 •*- r 2 2 + Krs 2 ) _ 

“o 2 ( f 0 2 * T o 2 •+• K^q 2 ) (5) 

Here K^q and &r© the radial velocity coefficients. Equation 

5 might he obtained immediately from, the relation 

°0 2 = c a0 2 ■+" c u o + c r0 2 - Uq 2 ( f Q 2 -+- T 0 2 ■+■ K^ 2 ) 

and analogously for c 2 . Since the radial components have very smai 1 values 

almost everywhere, so that their energy is small in comparison with the energy 
transfer in the stage, the difference ^ 2 “ mQ 2 ^ 2 is usually very «m«n 

and the ^ terms occurring in Equation (5) can practically always he neglected. 

For homogeneous stages, which are primarily considered in the following, 
fo = 811,3 To = ^2* so that 

V = 2 K 2T 1 - T 2) + (1 - V*) ( f 2 2 4- ti) (6) 

where K^ g 2 is neglected. If it is assumed that the stream surface is 

conical and that the axial length of the stationary blade annulus is the sama 

r (1+ mo)/2, so that 

- mo 2 ) if2 2 ) 

(7) 

Finally, if the stream surface is cylindrical, so that the stage element is a 
normal element, then 

y z A = 2 rci - t 2 ) - 2gj (8) 

where E p = — T 2 is ’ tiie circulation coefficient introduced 

by Keller. Naturally 1// = A only in the frictionless case. 

The distance from the blade grid at which the velocities and velocity 
ratios are to be taken has not yet been specified. However, this has little 
significance, since the energy ratios always remain the same if 0 and 2 
are approximately corresponding locations - for example, if they are both 
taken in the middle of the axial clearance space between the stationary and 
the moving blades'. 

2.3 Number and Choice of the Independent 
Characteristics of a Stage Element 

* The "characteristics" of a stage element are those quantities such as 
angle, degree of reaction, velocity ratio, etc., which characterize the general 


as that of the rotating blade annulus, then m^ 

2 


= 2 


fi-pj 


Vi -r 2 )+d 


13 


2321 







flow pattern and "behavior of the element "but not the special shape of the 
"blade profile. A stage is considered as determinate, or fixed. If all these 
magnitudes are given, "but it Is not implied that the geometrical shape of 
the "blades is fixed. Bather it Is mean'- V.t all the conditions are known 
which determine the proper "blade profile, or for which the "blade profile must 
"be suitable. 


Particularly characteristic of a stage element is the curvature of 
the two surfaces of revolution, infinitely close together, which "bound it. 
However, the effect of the details of the shape of these surfaces is vanish¬ 
ingly small* and it is simply necessary to know r Q , r-jy and r 2 . The 

stream angle in the meridian (longitudinal) plane is neglected, as is the 
direct effect of radial velocity components. If dimensionless numbers are 
employed, ihq and mp are the two numbers by which the shape of the stage 
element in the meridian plane is characterized. 


Further characteristic quantities, or numbers, which in the general 
case can be entirely independently chosen, are the four stream angles, 

C^l> /3±> and /^2* If these have been selected, then if 2 . 
is fixed by c(. i and since selection of absolute and relative 


stream angles at a given point is equivalent to fixing the ratio of axial 
velocity to tangential velocity. At the exit from the stationary blade grid, 
and at the entrance to the moving blade grid, these ratios are then com¬ 
pletely determined. 'At the entrance to the stationary blade grid and at 
the exit from the moving blade grid, since &- q and /&2 are 8 iYen > specifi¬ 
cation of a single velocity relation at each section such as and 

respectively, is sufficient to determine the relations. 


It follows from the foregoing that in the general case a stage ele¬ 
ment is determined by eight quantities or numbers, and that these might be 

<5^0 f Cf'lf /$1> /^2 1 *fof m 0) m l* It is, however, clear 

that the element might equally well be characterized by other quantities, 
which in the general case must always be eight in number. 


For the general element of a homogeneous stage, since 0^ 2 = cXo, 

the axial velocity ratio <^ 2 is determined by the four originally inde¬ 
pendently chosen angles, tod since further s (j ? 2> in this case only 


six quantities can be independently chosen. In most practical cases, as will 
appear later, the number of quantities which can be independently chosen is 
always less. For conical stream surfaces one of the m’s drops out, so that 
only five quantities remain which can be independently selected. For normal 
(cylindrical) stage elements m drops out entirely, so that only four 
quantities remain. If cylindrical or conical stream surfaces are assumed, it 
is also practically always assumed that all elements of the stage have this 
simple shape. In this case the four angles /$2. 


all be independently selected because these would determine 


U fi 


and 




and the determination would in general be such that these ratios would not be 
compatible with the continuity equation if the assumption Just mentioned is 
made. It Is possible, therefore, to select or independently specify only three 
angles or three equivalent quantities. 


The following is then the picture for the elements of a homogeneous 
stage. In the general cape such an element Is determined hy six quantities; 

for conical stream surfaces by four quantities, and In the case of a normal 

element by only three Quantities. 

It is to he noted at this point that the introduction of the continuity 
equation into the analysis introduces a complication arising from change of 
density. In order to take account of this effect a Mach Humber, defined as 
M = \v>/a (" a " equals velocity of sound at the conditions existing at point 2), 

is introduced, and the further use of the continuity equation would he depend¬ 
ent upon a proper value of M, which would he another independently specified 
quantity. This complication is neglected for the time being, hut a correction 
for it will he introduced in the later analysis. 

The necessary number of independent quantities has now been determined, 
and the question must he considered: which quantities are most convenient for 
use in characterizing the stage element? The angles have the advantage of 
being closely connected with the shape of the blade profile, hut they have 
very serious disadvantages. First, the angles hy themselves give only a very 
vague picture of the physical relations in the stage. Secondly, if the analy¬ 
sis is begun hy use of the angles, the equations became very complicated and 
physically unreal (as, for example, hy the necessity of using sine, tangent, 
etc.). The fundamentally simple relations which actually exist do not appear 
clearly in such a treatment. 

It is evident that it is possible to select the m agnitudes charac¬ 
terizing a stage element in many different ways, and that it is not possible 
in general to say which ways are best. In some problems a certain selection 
may he much better than in others. However, it can he shown that there is a 
certain group of quantities which it is practically always desirable to use 
in beginning an analysis, and these will be indicated in the following para¬ 
graphs. 


In all cases except that of the cylindrical homogeneous stage ele¬ 
ment - that is, the normal element - it can be shown that calculation of the 
other data pertaining to the stage element necessitates very complicated 
equations if a start is made with any other quantities than those which were 
used in equations 2.2 (4)-(8) for calculation of A and \fJ • ibis comes 

about because of the complicated kinematical relations. Under some circum¬ 
stances it nay be desirable to select ^ or &b one of the independent 

quantities, so that one of the other quantities would drop out - for example, 
either 'V i or Tq. However, in general V 0 , T^, Z2* f 0> fl* 

(j} g, idq, % are to be considered as the independent quantities charac¬ 
terizing a stage element. For a homogeneous stage, Z 0 and drop ouu. 

For a conical stage element, f x and ^ are not required, and if the 
stage element is cylindrical, b)q also drops out. 

Using these data, A and ^ are calculated from 2.2 (4}-(8) ; 

all stream angles and velocity coefficients are determined by simple geo¬ 
metrical relations which are evident from the velocity diagrams. An im¬ 
portant characteristic of the stage element is its dynamic degree of ire- 
action 9 which, from 1.4 m, is defined as 


0 = h "ad s Ap" (1) 

had AP 


15 


I 







where 


Ap 


f 


and 


A p" 


2 _ 


• -V { 


(T a -i) -i- ft 


- m i 2 [ < T l - i) 2 + f i 2 


Substituting these expressions in (l), and substituting also the value of 

V from 2.2 (5), neglecting , the value of 6 for the general 
stage element is 


e = 


(T g -l) a + 2 -»\ 2 [ - l ) 2 -H fl 2 ] 

2 <"i 2 ^ x *T a ) + ( f 2 2 + T 2 2 ) - ^ a ( f Q 2 + r 0 2 ) 

For the homogeneous stage element, 


( 2 ) 


0 = (T 2 -l) 2 -h ^ 2 g - m 1 2 [(r i -D 2 4- <p! 2 ] 


(3) 


W 


2(21! T i - t 2 ) + (1 -hJq 2 ) ( ^ 2 2 -f_ T 2 2 ) 

For the normal stage element (hJq - m-L = l and ^ i = If q ■ f ) 

B -- (T 2 -i ) 2 - (Ti-i)g 

a( r x - x a > 

If it is assumed that ^1 = 2 = # this is equivalent to 

considering the fluid to he incompressible. This must always he. assumed if 
a homogeneous stage is considered to he made up of normal elements. From 
2.2 (8), X ! - ( A/2) i- 'UQt and if this is substituted in (4) there 
results after a short algebraic transformation 


o = i - jl - r 

4 2 


(5) 


As 4 (i - a -r 2 ) 


( 6 ) 


16 


2321 







This very simple equation is fundamental for the further treatment of the 
normal stage element and particularly for the common treatment of turbine 
and compressor. This equation will now he written in a form which is 
later extensively used. 

A = Ml - p - 'X, 2 ) (7 


In this equation p is the kinematic degree of reaction. At first sight 

it appears from comparison of Equations (6) and (7) that it is unnecessary 
to introduce this quantity, since it is apparently merely substituted for 

0 . The reason for replacing 0 by p will become clear in the next 
section. For the present p may be considered to mean the same thing as 0 

It is evident that p , (^ , and 'V 2 might well be selected 

as independent characteristics for the normal stage element. For calculation 
of compressors it is often more desirable to select A , ^ , TT 2 or A , 

y . In the case of turbines it is often desirable to use A > p , 

°r A , <X x , /S z , 

2.4 Determination of the Dependent 

Characteristics of a Stage Element 

From the independent characteristics originally chosen, all the 
remaining (dependent) characteristics may be determined, particularly 

A , y/ , Q (cf. 2.3). The equations so obtained, however, possess'no 
very noteworthy interest. For a normal stage element, the relations become 
very simple, and in this case a very interesting fact appears. 

It is well known that in the case of a plane grid which friction- 
less fluid approaches with a velocity c^, and which it leaves with a 

velocity cj_, the force acting on a blade is perpendicular to the vector 

0 ^ + c 1 . Therefore the velocity c ^ s l/2 (c^ 7 - c^) has in grid 

theory, and consequently in the theory of turbomachines also, a special 
significance. If, for example, the case of the rotor blade grid of a normal 

stage element is considered, then w ^ - l/2 (w^ 4- ™o) is this "mean 

ideal stream velocity". If the vectors w^ , ^ are drawn from the 

same point, then it is clear that the end point of the vector w 00 ‘ bisects 
the line Joining the end points of w^ and Wg . 

Consider now the velocity triangles shown in Figure 4, in which the 
lengths of the lines indicate velocity coefficients rather than the velocities 

themselves. K„ and K are shown in accordance with the previous 

—C c&_ _ _ 

paragraph. Since the line EG is equal to AC, and AC = 'C ^ - "U 2 * 

X x 

and since, from the definition of K , the line FG - EG/2, 

ro = ~ (- X 2 ) - A 

2 T~ 


P > 

°C x 


17 


2321 





Since DG = 1 


1 


- Ta, 

HF ■ DG " ro = 


T 0 " ^ 

2 “IT 


Since, from 2.3 (7), ( ^A) - 1 ~ P ~ T 2 lt follovs immediately 


that 


- £ Tangential Component of X^ gQ ’] = DF = JO 


( 1 ) 


In the same way it is evident from the velocity diagrams that 

f Tangential Component of ^ ] = HD s 1 - p (2) 


The minus sign in Equation (l) expresses the fact that the quantity in the 
brackets is itself negative, since the rotor velocity is oppositely directed; 

In 2.3(7) the "kinematic degree of reaction" /) was Introduced 
instead of the "dynamic degree of reaction" 0 previously used. The 

difference between the two was not apparent at that time, but it has now 
been shown that f has a very simple geometrical significance in the 

velocity diagram - that is, a kinematical significance. 

All the previous development has been based upon frictionless flow. 

If losses had been considered the relations would have become complicated. 

In particular, 2.3 (6) would have been more complicated, and the simple 
geometrical meaning of the degree of reaction Just pointed cat would not 
exist. 


The kinematic degree of reaction p is now defined as that quantity 
whioh is given by Equations (1) and (2) (which pre essentially equivalent) of 
this section. Then 2.3 (7) will always hold even for flow with losses. 

In this case, however, p is no longer in general equal to the ratio 

h"adAad > tod is therefore no longer identical with' Q , but has, except 
for extreme oases, approximately the same value. For frictionless flow 
through a normal stage element p and 0 happen to be identical. For 
stage elements which are not normal p is not defined for reasons which aro 
explained In Chapter IV. 

It is apparent, then, that p is a very useful quantity for charac¬ 
terizing a normal stage element, because on the one hand it leads to simple 
equations, and on the other hand it possesses a clear physical significance; 
namely, it is the degree of reaction which the stage elexaant would possess 
with frictionless flow (and which in practice it approximately does have). 

The following equations can now be written 


18 


2321 





0 oO 


2 

K 


2 ^ ~ P ^ y 


1 - P 

V (3)00 



y? 2 -#- p 2 # 


°tg/£o 


n. 


(5) ( 6 ) 


If P > <P > to aro selected as independent characteristics of the 
normal stage element ( as will "be done consistently hereafter), the following 
equations are obtained, as may be verified from the velocity diagrams. 


*1 ■ 2 (1 -p) - T a (7) 

‘6 <<]. : - t- _ (8) 

ti 2 a -p ) - r 2 
ots oC o = '£ 3 . (9) 

f 

^ Ax . L ..Tji. = t 2 +1 - 2(1 -XI (10) 

l 

* 6/^2 = _(H) 

1 " *r 2 


if X, <p , r 2 or A , y<? , if y are selected as independent 
characteristics, then by solution of 2.3 ( 7 ) for p or TT g, respectively, 

the standard case is obtained where P > , X 2 are given. If, as is 

often desirable, in the case of turbines p , <?C 31,6 S 0 l ec ted, then 

X" 2 iB found from 2.3 ( 7 ) and p from ( 8 ), so that the standard case is 
again obtained. Values of p , <p , 7T 2 are obtained from A , cK. 

2 1>7 means of the following equations 


* -(4 

t | 

j tg 4 2 

tg <x 1 

(12) 

2 > 

(ft 

ro 

+ tg oC x 


= 


~T T 1 


(13) 

7 

ctg 


ctg fi 2 


r 2 


1 


ctg 



<*ixt g/ 3 2 -*-i 


( 1*0 


The "velocity ratio" (u/ci) which is often used is connected with these 
characteristic quantities by the following equation 

* Editor's Note: The symbols "ctg" and "tg" which appear as in the 
original text denote "cotangent" and "tangent" resoectively. 


\ 


19 


2321 

















sin ^ 

~T~ 


l/tMi-/))! (15) 


2.5 General View of the Normal Stage Element 


The equations of Section 2.2 up to 2.4 were derived on the basis of turbine 
phenomena. The relations are completely general, however, so that they are also valid 
without change for compressors. It must only be borne in mind that the pressure coef¬ 
ficient and the work coefficient /\. become negative, since a compressor is a 

turbine which does negative work. If, however, the calculations are made with positive 

A and , it is merely necessary to change the signs in all equations where 

these quantities enter; otherwise everything remains unchanged. The following should 
also be noted. The singles which enter into the formulas are to be measured as shown 
in Figures 5 and 4. The calculation is then absolutely uniform for the turbine and 
compressor. If one type of machine alone is considered, however, it is possible to 
carry out the entire calculation using only acute angles. If the corresponding supple¬ 
mentary acute angle is substituted for an obtuse angle, then in the formulas the signs 
of the cosine, tangent and cotangent must be changed. 

Through variation of p , p , and a H usual normal stage elements 

may be included in the treatment. Moreover, since in the case of axial machines the non¬ 
normal elements are always very similar to the normal elements, this permits a general 
consideration of all possible types of blading for multistage axial machines. The axial 
velocity coefficient (or flow coefficient) is not important insofar as the general type 
of blading (from the standpoint of its operating characteristics) is concerned. There¬ 
fore a very general survey is possible merely from the variation of p and alone. 

In Figure 5 such a survey is given schematically for possible normal stage elements with 

P varying between -0.25 and -f- 1.25 and with 'J ?g such that A varies 

between + 5 and -2. The compartments enclosed by heavy lines correspond to T* 2 = 0 

and therefore to axial discharge. Those compartments above correspond to cases where 
'Z 2 * B lass than 0, those compartments below to cases where 'Z £ is greater than 0. 

The compartments corresponding to cases of little practical interest are not filled in. 


The stage element just described must be thought of as one of the intermediate 
stages of a multistage turbine or compressor. It is evidently immaterial whether a 
stationary blade grid and the following moving blade grid are considered as a stage, or 
vice versa. All stage elements may be so specified that the moving blade grid follows 
the stationary blade grid. In the case of blading where, in the commonly understood 
sense, the stationary blade grid follows the moving blade grid, it is usually to be con¬ 
sidered that intake and exit guide vanes are added to the grid blading proper the shape 
of these vanes (as can easily be verified) differing from that of the other stationary 
blade grids. 

Figure 5 hardly requires explanation. However, the most important types of blading 
used in practice may be pointed out. 


Type of Machine 


P-. 

0 

A = 4 

Impulse turbine 

p- 

0.5 

A = 2 

Non-condensing turbine 

f- 

1.25 

/A a -1 

Axial compressor with the 


stationary blade grid, in which 
the fluid is accelerated, pre¬ 
ceding the moving blade grid. 


20 


2321 









Type of Machine 


P, X 


>> 

II 

H 

A 

P = 0.75 

A : 

pz 0.5 

A = 

P = 0.75 

A = 


Axial compressor with the 
stationary blade grid following 
the moving blade grid and acting 
merely to deflect the fluid. 

As in preceding case except with 
diffusion in the stationary blade 
grid. 

Axial compressor with diffusion in 
both stationary and moving blade 
grids, the blades of both being 
identical. 

Kaplan turbine, approximately. 


2.6 General Considerations Pertaining to the 
Stage of Finite Radial Length 


In a stage of finite radial length there is, in the general case, a very compli¬ 
cated flow. The basic analysis and special research in this field are taken up in 
Chapter 71. In the general case with frictionless flow there is in the various stage 
elements of the same stage a different increase or decrease of pressure energy per unit 
mass and also a different amount of work done. 


It is possible to use for complete stages the same characteristic numbers as for 
stage elements. They are referred to the tangential velocity u^ corresponding to the 

principal stream surface; that is, to the tangential velocities associated with the 
points A, B, C, shown in Figure 1. In general, it is sufficient to use the charac¬ 
teristic numbers referred to the principal stream surface as the characteristic numbers 
applicable to the stage, so that y g = y H , A S = Ap - 0 H> 

etc., where the subscript S indicates the characteristic number for the stage, and the 
subscript H indicates the characteristic number for the principal stream surface. 


In principle it would be more correct to use integrated mean values, and the 
corresponding integral expressions are easily obtained. For example, the pressure coef¬ 
ficient \j/ is obtained as follows. In any stage element corresponding to any 

arbitrary radius r 2 (cf. Figure 1), the increase or decrease of pressure energy per 
unit mass is equal to Ap - ^ ixg 2 \lj . The mass flow is 


2 7T r 2 ar 2 


2 'n T Z LI 2 if 2 u 2 dr 2 • 


It is now necessary only to write the total mass flow per unit time obtained by integra¬ 
tion and to divide one expression by the other in order to obtain.the total pressure 
energy change per unit mass. If the quotient is then divided by l/2 u ^ ^ f the 


pressure coefficient 



I 


J. 

Z 



W a is obtained. 

Y (»s/z) 

f i U if r z/ t i fz u z <lr z 

( D„k) 

fe/2 ) 

(3W4 


a) 


21 






If the radius ratio y 2 = z*2/ r n2 is introduced into this equation, and if the 

tangential velocity is expressed in terms of r 2 and the angular velocity, this equation 
simplifies to 


V 






(\a d( ji 

‘hints 

(2) 


This equation has "been written as an equality for "both y Q and A s , "because 
it holds equally for each. The upper limit of integration is Y 2 r r s/ r n * In 
an entirely analogous way the following equations may "be derived. 


fs = 




7 


(3) 


= 


0c = 


ly z - 0 fs ITh 

£ 


1 

(*) 


fas ^ 


2.H 


(5) 


Itx (3) and (4) the subscripts 0, 1, 2 might be introduced, in which case the equations 
would be valid for the corresponding plane’s (compare Figure 1). The derivation of 
Equation (3) was such that the total flow is given by 


o r nips if H (6) 

nhere ■&. ls the annular (free ring) area. Equation (h) le eo derive! that the 
rotational flow B (total rotation of the through flow) is equal to 


B " 6 ^ r H Ts ( 7 ) 

Equations (6) and (j) might also be written for planes 0, 1, 2. 

Tw . Possible, and often desirable, to replace all the actual elements of a stage 

by normal stage elements. In Figure 1, for example, the normal element EF*G* cor- 
responding to th$ stage element EPS is shown by dotted lines. It coincides at the exit 


22 


2321 









of the stage with the actual element and is equivalent to an element of an incompressible 
fluid with density equal to the actual density at point E. Quantities referred to these 
equivalent normal elements are denoted by an asterisk. For the homogeneous stage, 
therefore, 



( 8 ) 



A* 


3 A 


w 


y* - y = \ k +(i + v* 1 ) 


(15) 


Equations (8) to (ll), inclusive, express the condition of constant flow. 

Equations (12) tc (l^), inclusive, express the condition that the rotation or whirl, and 
therefore the work done, shall be equal. The requirement of equal pressure change is 
expressed by Equation (15)* 

Use of the equivalent normal element and assumption of frictionless flow are not 
tantamount to assuming that the pressure coefficient and the work coefficient are the same; 
the pressure coefficient is defined as equal to that of the actual element. The right- 
hand side of Equation (15) is valid in the given form only for frictionless flow. All 
other equations are valid generally. 

These equations have, in particular, the following very practical application. If 
it is necessary to develop some relations concerning a certain homogeneous group of stages, 
it may be considered that the actual stages are replaced by stages made up entirely of 
identical normal stage elements. 

Only as the last step in the solution is the actual flow calculated from Equations 
(8) through (15). 

If a stage is not homogeneous, it can be replaced by purely cylindrical elements 
which however, are not normal elements because T 0 is not equal to Tg * 

following additional equation is then available. 




(16) 


23 


2321 






and Equation (15) is to be replaced by 



+ + K 


r& 


)- 


2 . 



i-Z* + 







i 2 / . ^ 2 


(17) 


Henceforth all stages vill be considered to be made up of cylindrical stage elements 
(in particular, of normal stage elements), since the results may be carried over to the 
actual flow by means of the equations given in this section. 


' 2.7 The Normal Stage 

If a homogeneous stage is made up of normal elements, the quantity r ( Z]_ ”*^2) 
for each of these elements must have the same value, since 


^e 

- 

A 

U2 

< 

II 

< 

l 

2 ** 2 (Z 1 

- *Z 2 ) > 




2g 

8 


and since by assumption 

h e 

is ■ 

the 

same for every 

stage element, 


- 

r 2 

( 

T 

1 - X 2 ) 

- const. 

( 1 ) 

If, in particular, 








r 2 

T 

1 

II 


(2) 


r 2 

T 

2 

= *2 


(5) 


and 

r ^ z kj ( 4 ) 


■where , kQ , kj are constants, then by definition the stage is normal. 

Equations (2) and (5) simply state that the lav of vortex flow, c u r = constant, 
holds both before and after each blade grid, while Equation (4) states that the axial 
velocity is constant along the radius. If the actual flow through a stage is such that 
it can be considered as made up of normal stage elements, these conditions in general 
will be fulfilled. If this is not the case, the purely axial flow cannot possibly be 
in equilibrium, and if the calculation is still made with normal stage elements, it 
must be considered that the actual flow is replaced by a theoretical flow in accordance 
with Section 2.6. 

Equations (2), (3 ), (4) fix all the characteristic quantities for the normal stage 
completely as a function of the radius as soon as they are given for a stage element, 
preferably for the principal stream surface. The relations are easily derivable and are 
given below. 




f 


( 5 ) ( 6 ) 

( 7 ) 


24 








2321 









( 8 ) 


'-/^O )*',-/&) 


*> = ( ct 3 ^ ~Pg) ct 3 


(9) do) 





(ID . 






( 12 ) 



It is evident that the equations are valid also If, Instead of r 


■h ' Ye’ Pe’ et0 -’ 


the values based on any other selected radius are used.* While the use of Equations 5 to 
12 is limited to normal stages, the following relation holds for every homogeneous stage. 



(13) 


It should he noted that p depends on the radius in a very simple way, and that this 
dependency is very marked. For example, in a stage with l/l^ = 0.2 if /O is 

zero at the inner diameter, Equation (8) shows that it Is equal to O.36 at the mean 
diameter, and to 0.556 at the outer diameter. 

With the help of Equations (5) to (12), a survey of all possible normal stages 
can be made, supplementing the survey previously made of all possible normal stage 
elements. From Equation (8) it can be seen that a very interesting law holds; namely, 
If the degree of reaction of a normal stage is unity at one radius, it is unity at 
all radii. 


2321 





Ill 


THEORY OF GRID TESTING AND LOSS COEFFICIENTS 


3.1 Notation and Assumptions 


In addition to the notation given in Section 1.3, which is used throughout the 
hook, the following special notation is used in this chapter. 


x,y 


Coordinates of a rectangular 
coordinate system, shown in Figure 6. 




Unit vectors in the system x, y. 


£0 = V: + v 0l 


Velocity preceding the grid 


£msec 



£l I u ll + ' v !bi 

£ 2 = ^ 

C = U_^ i 4- v, j 

— o O oO — ~ 0° X- 


Velocity immediately following the 
grid; see 3.2 (2) 



Velocity far behind the grid 



Mean ideal 


stream velocity 


msec 



£0 I (£</ ^ 

£l = (£i/c^ ) 

£2 - ^—2! C o° ^ 

% 

A U = Au/c 


u oi + v oJ. ^ 

tV x i l 

^ l u 2~ u oj/ c oo 


Characteristic 

Velocities 

Deflection coefficient 


E Mechanical energy content 




r 2 

-2l 




sec / 

A 

% 

Total energy loss of the grid flow 



r 2 

_21 




sec 1 

A 

e p 

Energy 

loss at grid profile itself 



r 2 

-2l 



l n 

sec I 

A 

*M 

Mixing 

loss on downstream side of the 



grid 

£ m 2 sec -2 ! 

A 

P 

Total change of pressure energy 



[m 2 

sec _2 j 


2 6 


2321 




s 


IT 


T 


S 


a 


S 


-v 


£ 


°< 0 , ^ 1 , ^2, Ot-gQ 

T 

TT 

$ 

if 

A 


Subscripts; 


Force acting on blade per unit width 


[kgnf 1 ] 

Normal force (y-component of S) 

[ kgm -1 ] 


Tangential force (x-component of S) 

[ kgm -1 j 

Lift component of S 

Drag, or resistance, 

[kgm _1 J 

"Pressure" acting on the curved ele¬ 
ment ds of a reference contour (see 
Figure 6), made up of the normal stress 
p (pressure) and the shearing stress CT 




component of S 


Angles between c^, c^, ^ 

respectively, and the x-axis. 

Loss coefficient (subscripts have same 
meaning as for A E) 

Pressure ratio as defined by 3.3 (12) 

Function defined by 3.3 (13) 

Function defined by 3*3 (l4) 

Function defined by 3.3 (15) and (4) 

* 


0 - preceding the grid 

1 - immediately following the grid 

2 - far downstream 


The conclusions drawn in this chapter are founded on the assumption that the flow 
through the grid is uniform and not varying in time. That is, only the uniform central 
section of the stream, which is not influenced by the effect of the walls, is considered. 

3.2 Measurements Required to Determine the Grid 
Characteristics for an Incompressible Fluid 

In Chapter IV the calculation of turbomachine stages Trill be treated on the basis 
of grid test measurements. This calculation must begin with quantities which charac¬ 
terize the properties of the particular grid used, and which are integrated values ob¬ 
tained from measured values. The calculation of a stage requires the specification of 
two quantities entirely independent of each other; one to characterize the deflection 


27 


2321 


produce^ In the stream flowing through the grid, and the other the loss which results. 
These specifications may he given in a number of entirely different ways. Note that 
it is not necessary that one specification must he given to fix the deflection and the 
other the loss. For example, if in accordance with the usual airfoil theory a lift 
coefficient ^ and a drag coefficient are specified, each of these influences 

both the deflection and the loss. However, it is always necessary to have two entirely 
independent sets of values specified. 

This section considers the problem of what measurements must he made so that the 
necessary data may he obtained to specify quantities completely characterizing the grid. 
Nothing further will he said here about the self-evident requirement of geometric simi¬ 
larity, or about the necessity of equal Reynolds Numbers, since these matters have 
already been widely treated in the literature. The requirement of equal Mach Number 
does not enter here since in this section it is assumed that the compressibility of the 
fluid does not have any appreciable effect. 

Figure 6 shows a plane blade grid; the rectangular coordinate system x, y selected 
js shown in the figure. The x-axis is taken "near the exit plane of the grid gg", mean¬ 
ing thereby that the distance b between the x-axis and gg is so small that the velo¬ 
city and the pressure along the x-axis have not had an opportunity to equalize* The 
finite blade pitch t* produces a poticeable effect along the x-axis, which is evident 
particularly in the disturbed region immediately following the exit edges of the blades. 

In order to solve the problem, two regions are considered, indicated by AEEF and 
DCEF> respectively, and the equation of continuity and momentum law are written for these 
regions. The curves DAF and CEE coincide with stream lines which in the x direction are 
separated by exactly one pitch, and therefore have exactly the same shape. For purposes 
of calculation a section of the uniform stream is considered which has a width per¬ 
pendicular to the plane of the paper equal to unity. The distance of CD from the plane 
of the grid entrance plane gg is arbitrarily taken (in the limiting case this may be 

zero). The distance of EF from the plane of the grid exit is sufficient to make the 
effect of the finite pitch negligible for t he dertree of accuracy which is here in question. 
Theoretically, therefore, the distance of EF from the plane gg would approach the value 
y z oo — 


The velocity of approach to the grid is defined as 


C 

c (x, y Q ) dx, (1) 

D 

where c is the actual local velocity existing at different points along the line CD. 
In the same way the mean velocity along the x-axis is given by 



_1 

t* 



(x, 0) dx 


( 2 ) 


Along the line EF - that is, far from the grid - it is not necessary to obtain a mean 
value by integration in this way, since by assumption the velocity is uniform here, sc 
. that 

£ 2 = £ (^ 2 ) ( 3 ) 


28 


2321 



In grid investigations the test setup must be such that the velocity of approach 
to the grid is uniform and free from disturbances, so that the flow is very similar 
to a potential flow. Since rot £ = 0, is independent of the selected location 

of CD, and therefore is independent of Jq . The continuity equation requires that 

x + t* 

vdx - const. - v Q (4) 

for every given positive or negative value of y, so that 

®o ; V + V : A 

can depend on y 0 only if the first of these integrals depends on Jq . However, if such 

were the case, it would be possible to define a region DCC*D* bounded by the stream 
lines DAF and CBE, and by two lines y = y Q and y - y Q * , which would be entirely 

on the upstream side of the grid and for which the circulation would not be zero. This 
however, would contradict the assumption that rot c = 0. 

On the other hand, in general, 

since the same analysis can be repeated for the area ABEF, except that in this region 
rot c^ is not equal to zero because of the turbulent region immediately following the 

blade. 


C 'r 

^ u (x. 


y o> 


ax 4- j. 


t 


U, 


y o } 


dx 


(5) 



The velocity on the downstream side of the grid (which corresponds to a given 
approach velocity) is of fundamental importance, because it characterizes the deflection 
properties of the grid. From what has previously been said, the distance from the grid 
exit plane g a at which measurements should be made can now be determined. If the 

distance from g a is so great that c is practically independent of x, the mean value 

along x will be in all case3 practically independent of y, since in this case the stream 
lines are parallel, and a condition of potential flow may be assumed. It would there¬ 
fore in itself be correct to make measurements at a relatively large distance from the 
grid, so that a value of characterizing the deflection properties could be obtained 

directly. However, from the standpoint of test technique, it would probably be better 
to make measurements near the grid, particularly because the model is always only a 
relatively short section of a theoretically infinitely long grid. 

It appears, therefore, that c^, cannot be exactly determined by calculation from 
£^ but must be obtained from a direct force measurement on the grid. This requires a 
consideration of the momentum relations in the regions ABEF and DCEF. 

Considering a region quite general in character bounded by a surface K, and as¬ 
suming that at each point of the surface a resultant pressure vector n is acting 
(normal and’ shearing stresses exerted by the surrounding fluid on the surface), and 
further, that the fluid inside IC is acted upon by the external force F, then the 
time rate of change of momentum J inside the region is given by 


£ do i-n y (c, n) c ds (7) 

" 00 



29 


2321 



In this equation ds is an infinitesimal curved element of K and n is the inwardly 
directed unit vector in the normal direction. The second integral expresses the flow of 
momentum into the region. If Equation (7) is applied to ABEF, then, since dJ = 0 

dt , 

(because the flow does not vary in time) 




(-v) cdx 


o (8) 


In this'equation the minus sign appears before the integral because, for the direction 
of integration in question, dx = -ds. The integrals along BE and FA are omitted, 
because they mutually cancel. The two component equations, which are represented by (8), 
are 



O' (x, 0) dx 



J u (x, 0) v (x, 0) dx -t*u(y 2 )v(y 2 ) 
A 


o \9) 



P (x, 0) dx - t* p 



v^ (x, 0) dx - t*v^(y 2 ) 


0 ( 10 ) 


The integrals along EF have been evaluated in these expressions. From the previous 
discussion concerning signs it is evident that the direction of integration must be 
borne in mind, and further that at y z y^ > r -pj_., since the pressure in the 

region of integration considered acts in an upward direction. The second integral of 
Equation (8) does not occur in Equation (9) because at the point y - y 0 no shearing 

stress (5” can exist, since the flow here is a quasi-potential flow with parallel stream 
lines and completely equalized pressures and velocities. 


If the plane of measurement is taken at the x-axis, the following quantities can 
easily be measured: u (x, 0); v (x, 0); p (x, 0). From the continuity Equation (4) 
it is possible to find v (y 2 ), which is equal to 


Vp s v(y 2 ) : v 


1 _ 

t* 


C 

r 

v(x,y 2 ) dx 
* D 


v 


1 

t* 



v(x,0) dx 


( 11 ) 


the value of which is the same for all values of y. In Equation (9), therefore, there 
remain as unknowns only Ug - u(y 2 ) and the first term of the equation - the shearing 

stress integral. However, since the latter cannot be determined by direct measurement, 
and since enters only in Equation (9), so that it cannot be eliminated, it follows 

that U2 is still not determined. This confirms the fact that C2 cannot be exactly 
determined by , which is equivalent to saying that in theory it is not sufficient 


to measure the velocity distribution immediately following the grid along a parallel 
to the grid exit plane. 


In order to determine c^ 
will be witten for region DCEF, 


, the momentum law, in accordance with Equation (7), 
just as was done for ABEF. 


j 


30 


2321 










c„ F 

-S 4- / p dx - J' 


£ dx - / £ dx 

E 


/ 


The corresponding component equations are 

°r 

u(x,y Q )v(x,y 0 )dx - t*u(y 2 )v(y 2 ) 


-T + 


7 1 


-M + 


D 

t 

'D 


P(x,y 0 )dx - t*p(y 2 ) +// 


v c dx - / (-v)cdx 

'E 


= 0 ( 12 ) 


v 2 (x,y 0 )dx - t*v^(y 2 ) 


(13) 


= 0 (14) 


_ Shearing stress integrals do not enter into these equations “because the flow along 

DC and EF can “be taken as potential flow, although there is superposed a certain amount 
of turbulence. If the flow is a potential flow, there can he no shearing stresses, even 
if the fluid has viscosity. In such a case a slight superimposed turbulent motion would 
result in no interchange of momentum, and therefore in no shearing stresses, so long as 
the disturbance was not so great as to change completely the character of the flow. 


Considering Equation (15), it will be recognized that £2 can be determined as 
soon as the tangential component T of the blade force is measured, since u(x,y 0 ) and 
v(x,y 0 ) are measurable and v(y 2 ) - v Q can be calculated from Equation (ll). The value 

of - u(y 2 ), “the only quantity still undetermined, can then be obtained from Equation 

(13). It is therefore apparent that direct measurement of the force or at least of its 
T component would be the most accurate method of determining the grid properties. 


Exact measurements of force in such a case are rather difficult to make, and are 
usually avoided. This is Justifiable, since in spite of the theory which has Just been 
given, in most practical cases Just as accurate results can be obtained without direct 
force measurements. Theoretically this is possible only if the shearing stress integral 
is zero, but this integral in general will have a very small, negligible value, as will 
be clear from the following analysis. 

A large value of the shearing stress integral can arise only from the effect of 
the disturbed region immediately following the blade. If a traverse is made through this 
region, it will be found that there is a turbulent motion with a certain direction of 
rotation. As soon as the point of minimum velocity has been traversed (approximately in 
the middle of the disturbed region), the direction of rotation will reverse. Since the 
shearing stresses have a direct connection with this rotary motion, the sign of the shear¬ 
ing stress will also change, so that the net value of the integral will be approximately 
zero. The forces arising from viscosity and turbulence are of themselves small as com¬ 
pared with the deflecting blade force; hence the shearing stress integral, which depends 
on the difference between the viscosity force and the turbulent force, becomes vanishingly 
small as compared with T. It is therefore apparent that in practice it is necessary to 
make direct measurements of force only in cases of very great disturbance - that is, when 
there is considerable separation of flow from the blade profile. 


If the shearing stress integral is neglected, Equation (9) becomes 


”2 



u(x,0)v(x,0)dx 


(15) 


so that everything is determined without force measurements. 


31 


2321 










Same further Important relatione may "be derived from the foregoing equations. 
Consider that the line CD is so far ahead of the grid that the flow may he assumed to he 
practically parallel - say at C*D*. In this case u(x,Y 0 ) = Uq and v(x,y 0 ) = Vq , 

so that Equation (15) may he written 


A u 



T 

/^t*v 0 


(16) 


Note that the sign of T is changed because in practice T is considered to he positive, 
even if its direction is that of negative x. From this formula the quantity A u, 
which specifies the defleotion, may he calculated, either in the case when T is measured, 
or regardless of T if velocities are measured without any measurements of force. The 
fact that the formula assumes that measurements are made at a considerable distance pre¬ 
ceding the grid is without significance, since a disturbance of the flow on the upstream, 
side has far less effect than a disturbance on the downstream side. There would, there¬ 
fore, he no great error in using the inexact relation 



u(x,y 0 )v(x,y Q )dx 




v(x,y Q )dx 


especially as it is possible to make better measurements at a relatively large distance 
ahead of the grid. In fact, it is almost always permissible to calculate the velocity of 
approach from the flow and.from the known direction of the approach duct relative to the 
grid (and therefore relative to the x-axis). The velocity is then determined both in 
magnitude and direction. 

Proceeding with Equation (14) Just as with (15), the following relation is obtained 


Ap 



N_ 

t* 


(17) 


The same remarks apply to this equation as to (l6). It is to be noted that p 2 (the 

pressure for downstream) is not the same as the mean value of the pressure along the x- 
axie. This is apparent from equation (10), written as follows 



The left-hand side of this equation is the difference between p 2 and the mean value of p 
along the x-axis, while the right-hand side is in general not equal to zero, but is given 



If v were merely a constant, then it is apparent that this expression would be equal 
to zero. Let 



( 20 ) 


32 


2321 










If of all conceivable functions v = f(x) vhich satisfy Equation (20), the function 
v a k (vhich corresponds to a value of zero for Equation (19)) is that vhich makes 
the expression 

t* 

1 r v 2 dx 

t* 

Jo 

a minimum, then (19) is demonstrated. But this is exactly the case, as can be shown by 
the calculus of variations, or as can easily be deduced directly. This means, as can 
be seen by comparison of Equations (18) and (19) > that P2 is greater than the mean value 

of the pressure along the x-axis. That is, there is on the downstream side of the grid a 
certain pressure increase ( a kind of pressure regain) the process having a certain 
similarity to that occurring in an injector. 

The process of mixing on the downstream side of the grid, which evidently results 
in an equalisation of the velocities, is a process of irreversible interchange of momentum, 
and therefore results in losses vhich are not negligible. The loss is equal to the amount 
of energy transformed into heat by the turbulent motion, and so is equal to the difference 
between the mechanical energy per unit mass entering the region ABEF along AB and that 
leaving along EE. From definition 1.4a the mechanical energy is 


For incompressible fluids P 
is therefore 




E ** c -f- P 
2 

. The energy loss (that is, the "mixing" loss) 


t* 


% 


v(x,0) 


t*v. 


2 , 

o (x,0) 


P(x,0) 


dx - 


•f P 2 


( 21 ) 


This is a very appreciable fraction of the total energy loss of the grid, vhich is 


t* 

^ e G ■ 1 / v(x,y 0 ) 

Jo 

0 (x ' y 0> _ 4 P(x,y Q ) 

- 

dx - 

2 

f2_ + 

2 2 

r 


-1 

2 

+ P ° 


2 

°2 4 P 2 

2 

(22) 


The integrated form of the equation is based on the same assumptions as (15), (l6), 
and (17). The loss ^E K in the grid itself (that is, in the blade passages)'is 

A e g . A ^ + A ^ (23) 

To conclude this section, a very instructive numerical example vlll be given. 
Suppose a grid as shown in Figure 7, into vhich fluid enters with a velocity c Q = 1 

perpendicular to the grid axis. At the outlet plane of the grid the stream angle 

oCi = 17°28' , so that sincCi = 0.3. Let the blade pitch t* = 1. The 


33 


2321 



























x-rod.8 Is taken exactly in the plane of the grid exit. The component ug of the dis¬ 
charge velocity Cq 1b given ^7 (15)# so that the shearing stress integral is assumed 
to he zero. By continuity, v 2 r v Q - 1# 

In consequence of the viscosity effect, boundary layers are built up on both sides 
of the blade. The thickness of the layer is S ± and the concave side, ^ r .on the 
convex side as indicated in Figure 7. Inside of these boundary layers the velocity pro¬ 
file has the shape of a rectangular parabola. 

The pressure energies Pq and P]_ are to be calculated, P 2 = 0 being 

arbitrarily assumed in order to establish a reference point. Furthermore, , 

A E K , A Ej^ vi.ll be determined and these energy losses converted into the correspond¬ 
ing loss coefficients ^ G> t K, ^ M, vhich glve ^ ratios of ^ E to the 

energy conversion in the grid (definition 3.5). Because of the simplicity of the as¬ 
sumptions, the integrals can be calculated very easily. 


P -P 


0 1 = max 

2 


’l = V - 


2 sin %C 1 [ 1 - i < W ] 

Y v z M)aj= i - 1 --L ( ii +<S r ) 


- 1 

2 


f 


1 - 1 


( 5 i +< Sr ) j 

r V 

^ = f u(x,0)v(x,0) dx : ctg gC 1 • f ^(1,0) dx : 


1 ( ^l rf r ) 

i - j (S ± +& T ) 


ctg oL 


2 2 
”2 + v 2 


U 2 * 4- 1 


G = r 0+jQ_ 


"2 


y 2 

\ = / v(x,0) c (x,0) dx 4. ^ 


- c 2 


1 - & (Si+6 r ) 


2 sin* oC 


[i - i +<f r )] 


+ r l 


P, - Co 


5 ^ 


2321 













The following values, for example, would "be obtained 


+ 5 r 

p o 




A E 

M, 

0.1 

5.424 

-0.0202 

0.1674 

0.1278 

0.0396 

0.2 . 

5.836 

-0.0408 

0.3656 

0.2860 

0.0796 

0.3 

2.296 

-0.0617 

0.6027 

0.4778 

0.1249 

to 


? M 


^ 2 

0.0283 

0.0216 

0.0067 

0.236 

17°08» 

0.0377 

0.0451 

0.0126 

0.218 

l6°49‘ 

0.0887 

0.0703 

0.0184 

0.207 

l6°30» 


Even though the results have been obtained with very much simplified assumptions, 
nevertheless they are sufficiently good to determine the order of magnitude of the values. 
From the table the following conclusions may be drawn. The mixture loss amounts to more 
than 20$ of the total energy loss. This percentage becomes appreciably greater far 
blades with exit edges of finite thickness. The pressure energy regain, which is given 
by P 2 "Pi - “Pi f is small in comparison with the total pressure energy change, but 

is significant in comparison with the total energy loss A ^ , since it is more than 

10$ of the latter. The process of mixing results in a small but perceptible decrease 
of the stream angle cL , since /^ 2 iB leBB (Xi* 

3.3 Discussion of the Compressible Fluid Case 

Experimental investigations of blade grids at high Mach Numbers, where the com¬ 
pressibility effect becomes Important, would require much more extensive test equipment 
than for similar tests at low Maoh Numbers, The power requirements, in particular, would 
became excessively large. However, the theoretical relations have been deduced. 

There are cases (compressor grids with high peripheral velocities) in which the 
total change of pressure occurring in the grid itself causes such a small change in density - 
that as a first approximation the change may be neglected. At the same time, however, 
the average velocity of flow is so high that the very high local velocities existing close 
to the profile have noticeable compressibility effects. Such cases can be treated in 
the same way as the incompressible case, so long as the processes occurring in the im - 
mediate vicinity of the blade itself are not those investigated. This is true because 
the compressibility effect is equivalent simply to a corresponding change of profile 
properties. Since very flat profiles are always in question, it is usually possible to 
use test data which were obtained with practically incompressible fluids and to calculate 
the lift component S a of the blade force from Prandtl’s rule (compare Ackeret, Helvetica 

physica Acta) 


q = C 

°a comp a incomp 


This formula, however, can be used only up to a value (c /a) = 0,7, approximately. 

While the case Just mentioned arises in connection with compressor blades, very 
frequently for turbine blades the case is, in a certain sense, exactly opposite. The 
total change of density in the stream flowing through the grid is considerable, but the 



35 


2321 










Ioc&j. excess velocities in the passage are relatively small, so that the local changes 
of density are not much greater than the average change. The stream line pattern of 
such a flow differs very little from that of an incompressible fluid; the flow is • 

merely more strongly accelerated without the effect appearing in the stream line 
pattern "because the greater acceleration corresponds merely to a decreasing density. 

The deflection properties of the grid for this type of flow are therefore practically 
the same as in the incompressible case. 

Such grids usually have the property that the discharge angle is nearly independent 
of the entrance angle, "because the pitch is relatively small. It metf - then "be assumed 
that "because the general character of the flow is not changed, the "boundary layer pheno¬ 
mena remain approximately the same also, and thus the loss is approximately the same. In 
this case calculations can "be made with grid data obtained in model tests with an in¬ 
compressible fluid, without introducing any greater error than is unavoidable in any 
case in such calculations. Almost always this is true when turbine blades are in question. 

For purely geometrical reasons, the imitation of the actual flow relations in a 
model is much more difficult for a compressible fluid than for an incompressible fluid. 

In a turbine stage, for example, the meridian stream lines diverge strongly when there is 
a large compressibility effect. In the model this would require a corresponding slope 
of the side walls. However, this would lead to very complicated relations in a model 
test which would make a correct interpretation of the model test data extremely diffi¬ 
cult, The side walls would exert pressure forces which would have components in a 
direction perpendicular to the grid axis, and these forces would depend on the pressure 
distribution in the passage. It is therefore desirable to forego testing with this com¬ 
plicated three-dimensional flow and to make tests with parallel side walls Just as with 
an incompressible fluid. Evan though the test made in such a fashion departs somewhat 
further from the actual case, nevertheless the flow relations are simple, and the results 
Obtained will be much more valuable since it is known Just what they really mean. 

In grid research of this latter type it 1 b sufficiently accurate to use the sama 
general analysis which was developed in section 3.2. The only difference is that the 
continuity equation assumes a more complicated form, so that the other equations are also 
more complicated. However, they are quite analogous to those of section 3.2. 

The Cauchy Number must be the same for-both model and full scale apparatus. 

0 =£ ( 2 ) 

a 

For the same ratio of specific heats k, this is equivalent to saying that the Mach Number, 
or the pressure ratio p 0 /p 2 , must be the same. 

It is well known that, for the flow of an incompressible frictionless fluid through 
a straight blade grid, the blade force S is perpendicular to the velocity factor o ^ - 
1/2 (°o+ £»), so that the scalar product 


— (OjQ 4- £ 2 ) s 0 ( 3 ) 

The component of force in the direction of £ in the case of flow with friction is there¬ 
fore to be considered as a direct resistance or drag force. It is of Interest to loiow 
what direction S must have in the case of 'a frictionless compressible fluid so that in 
this case also the resisting force can be ascertained for flow with losses. This direct¬ 
ion is given by 


A 


s - (Oq + A 22 ) z 0 


00 


is a scalar factor to be determined. Considering again the region D*C#EF, Figure 6 , 


3 6 


2331 





\ 


the momentum lav gives as the equation of equilibrium 

-s -f t* [(p 0 -p 2 ) j1 + A) v q2d “ 2 V 2^2J - o ( 5 ) 

This corresponds exactly to the relation 3*2 (12) except that the integrated form is given 
and that two different densities yU q and now en '^ er * If the continuity equation 
is written for the same region, it follows that 

M 2 V 2 = y^0 v 0 (6) 


so .that ( 5 ) may also he written 

§ = t* £ (P0-P2) 1 +/A 0 V 0 (°D“£2)j ( 7 ) 

This expression for S is now substituted in (4), giving 

t* £(P 0 -P 2 ) 1 * (£o + ^ ° 2 > + y^o v O (£0"22)* (So+Ao^J- 0 (8) 

Multiplying out the scalar product, 

(P 0 -P 2 ) (v 0 +ylv 2 ) <?r 0 [ o 0 2 — A o 2 2 4- 

(A - 1 ) £o> Cg ] : (p Q -p 2 ) (v Q -h A v 2 ) + 

/< 0 T 0 [ C Q 2 “ A C2 2 + (A - 1 ). c 0 c 2 cos (tf< 0 -^ 2 )J = 


0 (9) 


From the laws of thermodynamics it is known that for frictionless flow 


2 2 
C 2 - C 0 


~T ~-1 


_ 2 _ = 

"0 




y^O 

A* 2 


1 - 


Po 


1 e 


( 10 ) 


p 0 ^ z 


P 2 


( 11 ) 


The latter equation would hold equally well if Vq and v 2 were specific volumes, 
since the y components of the velocity are proportional to the specific volumes. Noting 
now that v Q - c Q sin and v 2 = c 2 sin (A 2 , and substituting ( 10 ) and 

(ll) in (9)> the following result is obtained. Let 


TT : 


p 2 


( 12 ) 


PO 


f M = 


■ft -1 


1 - TT 


><-l 

AT 


(13) 


37 


2321 
















tfr ( J, <* o, <*2> 


2 


-i m 


Tf 


- 2 

£ / sin < 3 C 




Then 


A= i -f 


i fx -irjfi + tt'-e] - $ 


(008^0008^2+ Bln^7C 0 8in(7( 2 )y^tl J 


(15) 


■ *[‘ 'Tl]lT^ 

(Ed.Note: This is a continuation of the denominator). 

This relation is unfortunately very complicated. If the pressure ratio TT" is nearly 
equal to unity, the total change of density is very small, and this expression must 
approximate very closely Equation (3), vhich holds for an incompressible fluid. Com¬ 
parison of (3) and (4) shows that 


lim A 
TT-^1 


(16) 


This can also he deduced easily from (15). The fraction on the right-hand side of this 
equation must approach zero when 7T approaches unity. Since, however, the denominator 
may he given quite different values for a given Jf hy choice of different (A Q and (A 2 

it is sufficient to determine whether the numerator alone approaches zero. Using (10), 
this numerator (called Z) can he written as follows 


Z = 


1 


1 '(f) 


% Pr 


y Ji 0 


Z 3 


PQ-Pg 

S* 0 


If (P 2 /PQ) approaches unity, 


1 + / fg 

PO 


1 

t 


so that in this case 


1 + 


- 1 
It 


- 1 
S 


- c f 


it £2 


- ^W) 


■fep, 


- (c 2 2 -o 2 )> 


ip. 


z-*(p 0 -P 2 ) - ^ (c 2 2 -c 0 ) 


38 


2321 



















In the limiting case considered, however, Bernoulli’s equation must hold, so that the 
right-hand term "becames zero and 



This proves (l6). 

The results derived in this section may be summarized as follows. For uniform 
flow of compressible, frictionless fluid through a plane blade grid, the blade force S 
is perpendicular to the velocity vector c^ -f- c^, as expressed by Equation (4). 

The scalar factor -A. is given by (15); it approaches unity when the pressure ratio 
p Q /p 2 approaches unity. This gives an additional proof of the relation expressed by 

Equation (3), which is valid for incompressible fluids. 

3.4 Deflection Coefficient 

The deflection properties of a grid are determined if the discharge angle 
is known for every entrance angle (/. Q , (compare Figure 8). The simplest method of 

specifying the deflection properties of a grid is therefore to give (X 2 as a function 
of <*0. Unfortunately, however, the angles enter in a very complicated way in the- 
equations for the calculation of the machine stages. It is usually desirable to 
characterize the deflection properties of a grid by the angle itself, if (X 2 not 

depend upon (Xq . Moreover, this is usually the case for grids which have such a 
large change of pressure that the total change of density also becomes appreciable. 

If consideration is limited to an incompressible fluid, A U is in general 
found to be a very useful deflection coefficient (cf. Figure 8). Let 


£0 

-._?£= u o 

i + Jo 1 = Uoi + V 

C oo 

(1) 


_ ^2 = U£ 

C o O c oO 

i +- la 1 » Uai + T 2 i 

( 2 ) 

with 

c = 1 

~ aCr 2 

(£0 £2) 

(3) 

Then, by definition 

Av | £2-0,, | 

: | | = j 

1 U2-U0 1 r 4u 

CoO Coo 

(4) 


Furthermore, as can be seen immediately from Figure 8, 


ct s eg = «o ^ (5) 

^0 

•* 

The deflection properties of a grid determined by experiment might therefore be expressed 
by the functional relationship. 


Au - f (ocj) 


( 6 ) 


39 


2321 






From the analysis of Section 3.3, it is practically always possible to consider the 
relations existing in the case of compressible flow through a grid Just as if the fluid 
were incompressible, either because only local changes of density are appreciable, or 
because the general relatione of the grid are almost the same as for an incompressible 
fluid in spite of an appreciable change of density throughout. Thus grid research can, 
in fact, always be carried out in the region of vanishingly small compressibility effect. 

• 

Finally, it may be noted that in general 


\ 


Au = 2 | sin ctg - ooB^X^j (7) 

which is self-evident from the geometry of the configuration. This equation is of 
interest for the reason that in the case where gi 2 is independent of oC Q , it ex¬ 
presses explicitly the functional relationship 4U - f )• 


3.5* Grid Energy Loss Coefficient 


In Section 3*2 the total energy loss A Eq of the grid was introduced (compare 
3.2 (22)). Just as in the numerical example given under 3*2, the pressure at the grid 
exit (strictly speaking, at a considerable distance from the grid) will be taken as zero 
fOfnt of the pressure energy; that is, P 2 ■ 0. The energy loss coefficient of the 

grid Y G is then defined as the ratio of the energy loss A Eq, to the mechanical energy 

content (kinetic plus pressure energy) of the fluid preceding the grid (all per unit mass). 
Therefore 





-f- 2P 


In an analogous way the partial loss coefficients Yk given In 3.2 are 

equal to the ratios Ae^/Eq and A^/Eg, respectively. 


5 g gives in every case a reliable picture of the quality of a grid. Two 
different grids are best compared by comparing their respective value of ^q.* There 
is no case in which the definition of ^ would fail. Unfortunately this advantage 
must be coupled with the disadvantage that the calculation of stage efficiency using Yg 

leads to a somewhat complicated equation. This might be avoided by another definition 
of energy loss coefficient - that is, by letting 


1 - Y 



2IV 


2 


In this case, however, Y would became equal to unity for a pure deflection grid 
(c 2 = c 0 ) since P 0 is not equal to zero because of losses. Furthermore, if c 2 is 

less than cq and Pq z 0 ("classical" rotor grid of impulse turbines), — cO 

An energy loss coefficient so defined would under certain circumstances have values of 
little physical significance, and it would not be possible to obtain from them any 
idea as to the quality of the grid. In the limiting case such a definition would fail 
altogether. 


40 


2321 











3*6. Grid Force Coefficients and the 

Loss Coefficient Derived from them. 

The introduction of grid force coefficients brings the theory of turbamachines 
into a form which departs very appreciably from current steam turbine theory. For 
machines with blade profiles similar to those of airfoils, a treatment based on air¬ 
foil theory was introduced sometime ago and has proved to be very fruitful. The break¬ 
ing down of the blade force into a component S & (lift component) perpendicular to the 

direction of c M and a component (drag or resistance component) in the direction 

c ^ has a physical significance which has already been brought out. This significance is 

entirely independent of the special shape of the blade profile, and holds without restric¬ 
tion for profiles with such strong curvature that they bear no resemblance to the shape 
of airplane wings. In such cases Sy for frictionless flow of an incompressible fluid 
is equal to zero, so that the component ^ which actually occurs expresses directly 
the effect of the friction forces. 

It is to be considered, therefore, whether or not the relations of the airfoil 
theory might not be used quite generally for the calculation of turbomachines, the only 
exception being that the force coefficients ~£ a and w would no longer, in general, 

have any relation to values measured with a single airfoil, but would have to be found 
from grid measurements. 

On the other hand, direct force coefficients for the tangential component T and 
the normal component N of the grid force might be introduced Just as soon as corre¬ 
lation with the measurements made on a single airfoil is abandoned. This reduces the 
equation for the efficiency of a stage element to a particularly simple form, although 
there is also the disadvantage that the physical relations are not brought out very 
clearly. Both types of force coefficients are introduced here, and the corresponding 
equations are both given, so that the equation which is most convenient for the particular 1 
case may be used. 

(a) Lift and Drag Coefficients, Drag-Lift Ratio 
(or Glide Ratio) 

The force S acting on the vane with a fluid considered incompressible is broken 
down into a lift component perpendicular to c ^ and a drag component parallel 

to c . If s is the profile chord (see Figure 6), let 
- oO 


£ 


S a = 

^■ S /T 

z 

(1) 

^w - 


(2) 

£ - 

*2 - Sv 

sf 

(3) 


These equations define the lift coefficient a , the drag coefficient ~~£ w , 811(3 

the lift-drag ratio £ (also called glide ratio). £ is a direct measure of the quality 
of the grid, so that if a calculation is made using airfoil theory, this enters in the 
place of ^ q. 

It can be shown by the momentum law and by simple geometrical relations that the 
following relations exist for the uniform flow of an incompressible fluid through a grid 
if the conditions which have been assumed can be considered to be sufficiently well 


« 

4l 


2321 



satisfied. The derivation of these relations has teen given frequently in the literature 
(compare, for example, B. Amstutz, Stodola-Festschrift). 


AP^a If! 

s 

(cos djt £ sin<^) 

(fc> 


t* 


AU = — 

2 

JL 

t* 

(1 ± £ ctg C^oo ) 

(5) 


Here A P is the total change of pressure energy in* the grid, which for inc empress it le 
fluids it is equal to | (pq-P 2• Th® upper sign is used for accelerating grids, 

the lower sign for diffusing grids. The double sign enters because w , and £ 

are considered always positive for both types of grid. Otherwise Equation (5) is merely 
one of the expressions of the functional relationship 3*^ (6). 

If an airfoil profile is used for the grid, then, as is well known, it is possible 
to make use of the aerodynamic measurements made on isolated airfoils, provided that the 
effect of the grid structure is taken into account by some approximate correction, since 
in actual multistage machines the blade pitch is always relatively small. Methods for 
the approximate determination of this effect have been frequently proposed. One which 
is very practical and which is fully reliable within the limits of accuracy in question 
is that of E. Weinel (Ing.-Archiv., Vol. V, Page 91). The method of calculation there 
given was developed for accelerating grids, but it can be extended without change to 
diffusing grids if the curvature and angle of attack of the profile are considered negative 
in the calculation. It is found in this way that the lift coefficient of a profile used 
In a diffusing grid is not in general larger, but actually is smaller, than for a single 
airfoil. This is in agreement with experience, and in opposition to the view sometimes 
expressed that theoretically an Increase of the coefficient should occur, but that in 
practice this is neutralized by reason of boundary layer effects. 

(b) Tangential and Normal Force Coefficients, 

Friction Coefficient 

The following equations define the tangential and normal force coefficients 


2 



Co* 

2 

(6) 


2 

C 0o 

T 

(7) 


It would be possible to select reference lengths and reference velocities other than t* 
and c 00 , but these give the simplest equations. It follows directly from 3.2 (l6) 

and (17) that 


A p 




( 8 ) 


A u 


*T 

2 sin OC^q 


( 9 ) 


k2 


2321 




These equations also give indirectly the loss, which does not enter explicitly. In order 
to construct from and ~^ T a number which will be a direct measure of the loss, 

the following procedure is carried out. For frictionless flow the equation 





( 10 ) 


must hold. Ibr flow with friction through an accelerating grid the coefficient 

must be larger than the value given by (10), and for diffusing grids ^ ^ must be smaller. 

Let 


so that 



~f T (ctgc*:^ ± & ) 


19- 



(n) 


This is a quantity which serves as a direct measure of the loss. ^ is called the 
friction coefficient. It is the difference between the actual and the theoretical values 
of ^ jj /, and therefore shows substantially by what amount the force N and the 

pressure difference p 0 -pg depart from the theoretical values. There is a certain dis¬ 
advantage in that "d" does not give a clear physical picture. It would be possible to 

construct from ^ and in some other way another loss coefficient, but diffi¬ 

culties would enter in this case also, as has been pointed out already for analogous case 
in 3.5. The definition as here given has the advantage that it will never fail in any 
limiting case which is practically possible. 

3.7. Relation between Loss Coefficients 

The most advantageous way of specifying the deflection properties of a grid is to 
specify A U, while the Iosb can be characterized by £ , or . Charac¬ 

teristics of a grid are then completely determined if A U and one of the loss coeffi¬ 
cients as a function of (X^, as well as the Reynolds Number and the Mach Number, are 

known. Usually, for the sake of simplifying the engineering treatment, the effect of 
Reynolds Number and of Mach Number must be neglected, although this is not always entirely 
safe. Very often turbomachines operate in that region of Reynolds Number where the effect 
on the losses is appreciable. In the case of axial compressors, the Mach Number is often 
of appreciable importance at the tips of the blades. 

AU on the one hand, and , £ , or ^ on the other, are quantities 

characterizing the g_ d relations on which the calculation of energy conversion and of 
efficiency of the stage element are based. As compared with these ^w, T, 

and ^ jj have relatively minor significance. In theory the three loss coefficients 
r^Q £ , and ax ' e equivalent to one another, with the single limitation that £ and 

are defined only when the flow through the model grid is sufficiently similar to that of 
an incompressible fluid so that the scalar product S ♦ c ^ - U, at least approxi¬ 
mately. It must therefore be possible to express each of these loss coefficients in 
terms of each of the other two. The formulas are given below. They are easily derivable, 
since only a simple formal mathematical transformation is required. 


^3 


2321 






££AU 


[i , n *<*> + £ °° s ^oo) 1 Au t sO + un'H*)] 


0 


__ <g z^zatl sm^oo _ 

I ±AU^cos<^o ±2T?si r '<X*o] + (-^)^ 


±%\ 

jj^AllCosc^ -r(^r) 

> a ] 


±aU| 

,n + (2 .-T<j) CO sJ 

!*^j 

r 1 * (wj 

I CCS <7(00 


&> 


<5 


_ # 

I + c'fcg^oa 1(7 z^c% ®C oo 


< 



•$ 6 [l± AOcosrf.'O+teit') J 

z(l~f G )&0 sin ctC^ 


ft) 

ft) 


f^l-t- etgV^,) 

t + £ dg *<«, 



Here again the upper sign is used for an accelerating grid; the lower sign for a 
diffusing grid. 

Strictly speaking, of all the grid characteristic quantities which have "been 
introduced, only 1s j ^ 1-1)8 definition, independent of whether the fluid is com¬ 

pressible or incompressible. All the others are based on the assumption that the relations 
which exist are those occurring with an incompressible fluid. It will be seen, however, 
that in spite of this fact, such quantities can be used in all practical oases to which 
the present theory is applicable. 

Of the various loss coefficients which have been introduced, _ has the advantage 

G 

of being easily visualized, because it is exaotly the ratio of the energy loss to the 
energy content of the fluid preceding the grid. Therefore ^ q is also most easily 
estimated if no measurements are available. € is easily visualized and estimated only 
in the case of airfoil profiles, but in this special case it is even superior to 

l^is not easy to visualize, nor can it be easily estimated, but it does lead to the 
sin?)lest equations. 

In the case of axial flow compressors, it is usually advantageous to use <£ , but 
in the case of turbines it is best to work with "S' or v . In order to connect the 

advantageous definition of ~£q with the simple method of calculation made possible by 
use of , Table 1 (see Appendix) was computed. This simplifies the use of equation 
(5), which may be written 


kk 


2321 








( 7 ) 



^>G 


i+^UcosoC^ * 



2 AU sin 


— ie _ f(au^) 
I - 'Sq 


The function F ( A TJ, G^ a0 ) is to be taken from Table 1. However, this method would 

normally be used only for accelerating grids, since for diffusing grids it is usually 
better to work with £ . 


3.8. Numerical Example. Data from Tests 
made with an Accelerating Grid. 


As a numerical example there are given below data from grid tests made in the 
Fluid Dynamics Laboratory of Gebr. Sulzer A. G., Winterthur. The wind tunnel, which is 
intended particularly for grid research, and the measuring equipment belonging to it, 
were designed by dipl. Ing. M. Fauconnet. 


Figure 33 (see Appendix) shows schematically the construction of the wind-tunnel. 

A double-entry Sulzer blower delivers air through a transition piece into a long conical 
diffuser whioh opens into an equalizing chamber. From this chamber the air is led directly 
to the model under test, and then through a nozzle-shaped opening where it is strongly 
accelerated. In the transition piece between compressor and diffuser there is a flow 
straightener (a net-work of baffle plates), in which the strong secondary whirl persist¬ 
ing after the air leaves the compressor is eliminated. Complete equalization of the flaw 
is accomplished in the equalizing chamber by means of baffling devices with ”straighteners" 
and perforated plates. This arrangement results in an extremely uniform flow to the model. 
The wind tunnel is otherwise similar to that whioh the Aerodynamics Division of the E.T.H. 
had shown at the National Swiss Fair in 1939. 

The grid on which the tests were made is shown in Figure 3^ (appendix). It is an 
accelerating grid with blades of an airfoil profile similar to MCA 6309 (described in 
MCA Report No. 460). Figure 3b gives all other pertinent information concerning the shape 
of the grid and the location of the planes at which measurements were made with a cylindri¬ 
cal impact tube. The grid width perpendicular to the plane of the paper was 400 mm. 

The grid actually consisted of nine blades, and not of only five as shown in Figures 33 
and 3h for simplicity.. The blades were of a light metal. It is to be noted that the pro¬ 
file angle shown in Figure (8°15*) is the angle included between the horizontal line 
and the profile chord to which all airfoil properties are referred by MCA (in particular, 
the angle of attack). 


It appears at first glance that there would be little or no application for a grid 
of this type in multi-stage turbines. It can easily be shown, however, that the con¬ 
ditions existing at the outer portions of the rotor blades of reaction turbines would be 
suitable for the use of such a grid. If the condition rc u a constant . is to be 

satisfied both before and after each row of blades, the flow angles would be such as to 
require grids of this type even for blades which are not unusually long. 

The test results are given in Figure 35 (Appendix). The variation of the velocity 
head ( c^, of the total pressure p -f- ( /2) c^ , and of the flow angle (X. 

over a distance equal to one pitch are given for each of the three planes of measurement. 
The plotted points are those taken near the middle of the group of nine blades. The 
curves should theoretically have a strongly periodic character, but this effect does not 
appear in its entirety because of the imperfect nature of the test equipment. 

The distance of measuring plane 2 from the grid in the direction of flow was so 
great that useful measurements could be made only at a middle portion extending over a 
distance equal to approximately two pitches. Measurements at different heights (perpen¬ 
dicular to the plane of the paper in Figure 3*0 show that the flow angle and velocity 
change very little along the height except, of course, in the region very close.to the 
walls. For a middle section 200 mm wide, these departures are so sma l l that they are 


45 


2321 









vi thin the limits of test accuracy. Since the test points were taken in this middle 
section, they reflect the properties of the profile itself. All test values are 
corrected to a total pressure preceding the grid of 70 kg m~ 2 (- 70 ^ h^). 

I'ram readings taken at measuring planes 0 and 1, the deflection coefficient and 
the loss coefficient may he calculated (neglecting the shearing stress integrals). The 
HflTrw values might he calculated independently from readings taken at measuring planes 
0 and 2, so that a check is possible. Using the equations given in this chapter, the 
following values are obtained. 

• 

Measuring Measuring 

Plane Plane 

1 2 _ 


p o 

kg 

-2 

m 


54.80 


°0 

m 

sec 


16.05 


”0 

m 

sec”^ 


14.26 


V 0 

m 

sec -2 


7.34 


o(- 

0 



27°12* 


P 2 

kg 

m -2 

O.36 


-0.40 

c 2 

m 

sec -2 

33.87 


34.02 

us 

m 

_2 

sec ^ 

33.03 


33.20 

V 2 

m 

_2 

sec 

7.48 


7.46 

<*2 



I2°46‘ 


12°40* 




0.0293 


0.0285 


The velocities are calculated for a density yU - 0.118 which corresponds, 

for example, to the conditions 20° C, 731 mm barometer. 

The agreement of the calculated results based on the readings taken at the two 
different planes of measurement is very satisfactory. Comparing v 0 and v 2 , it is 

apparent that the stream contracts slightly sidewise; that is, the boundary layer at the 
side walls becomes somewhat thicker. However, this effect is very small, since the 
difference between v 2 and vq for the two cases is only 1.7$ and 2$, respectively. A 
further small "disturbance effect" is apparent from comparison of the tvo values of p 2 . 
It is evident that there is a certain decrease of pressure on the downstream side of 
the grid. 

It is to be noted that near the grid a somewhat greater loss is measured than at 
a greater distance, in contrast to what would be expected. The difference, however, is 
very small. In order to estimate correctly the accuracy of the measurements, the values 
of 1 - "§ q. (not muBt be compared. These two numbers are 0.9707 and 0.9715, 

which differ by only about 0.8$. It is quite apparent that a greater accuracy is hardly 
to be expected. 


46 


2321 




From this example it may "be concluded that in the case of efficient grids (as 
accelerating grids usually are) the losses are so small that the error Introduced hy 
imperfection of the test equipment is somewhat greater than .that which arises from 
the principal theoretical uncertainty of the method (neglect of the shearing stress 
integrals). Therefore velocity measurements are sufficient, and it is not necessary 
to resort to direct measurements of force. This is very fortunate, since experience 
has shown that it is extremely difficult to make sufficiently accurate measurements of 
force in the case of these efficient grids. A direct force measurement was attested 
in this case, hut it was not possible to obtain any reliable results. The method of 
force measurements may, however, be advantageous in certain cases. 

A further conclusion which may be drawn is that the flow at measuring plane 1 is 
so far equalized that the "exit loss" from the mixing process on the downstream side of 
this plane and the "pressure recovery" are both negligible. This may be demonstrated by 
calculation. 


The assumption is now made that the slight contraction of the stream (which makes 
itself evident in the computed results by the difference between vq and v 2 ) is negligible 
in the following sense. If for the same angle oi. q arid the same total pressure (p 0 -f 

c 0 2 ), Cq were large enough so that vq = V 2 (no contraction), then neither 

the exit angle (K 2 nor the grid energy loss coefficient “^q. would be markedly in¬ 
fluenced. For the very small contraction which is actually present, this assumption 
may be safely made. With the value of co bo fixed, the results of the grid test may 
be presented in the following final form, based on the mean values of the readings taken 
at the two planes of measurement 


II 

o 

o 

16.33 

m 

sec - *" 

v z 

7.47 m 

sec - '*' 

11 

2° 

14.52 

m 

sec”- 1 - 

C oo = 

24.97 m 

sec” 1 

II 

27°13’ 



it 

* 

17°25’ 


C 2 = 

33.92 

m 

sec - ''" 

A u = 

U2-Uq 

= 0.745 


The values of 


y^2 = 33.11 m sec - *- 

2 z 12°43* 

C and & may be determined from 



0.0290 


£ z 0.0107 

1?- = 0.124 

The lift coefficient may be calculated from 3.6 (5). Its value is ~€ a « 1.245. 

Therefore the angle of attack of the profile referred to Coo is 4°20*. The fact that 

a profile used in a grid arrangement with such an angle of attack may have a lift coeffi¬ 
cient of 1.245 is in itself not to be wondered at, since the direction of c,*, has been 

fixed by purely theoretical considerations, and it is by no means the same as the actual 
direction* of approach of the stream. It follows that for grids with such closely spaced 
blades it is not permissible to use test data obtained with a single isolated airfoil. 


47 


2321 . 



In conclusion, the following should also he noted. In turbine design there is 
often used a simple rule of thumb formula which states that the sine of the exit flow 
angle is given by- 


sin oi p r actual passage width 

blade pitch 

By "passage width" is understood the radius of a circle which has its center on the exit 
edge of one blade, and which is tangent to the back of the adjacent blade. In the case 
of the test grid the passage width was 25.5 urn. (see Figure 3*0 * The blade pitch was 
105 mm. According to this simple rule, sin 2 ■ 0.2429, while the test results 
showed that actually sin <X~ - 0.2202. The effective flow cross-section area of the 

grid is therefore less than that corresponding to the simple rule; the ratio of the 
actual to the supposed area is 0.907* Expressed in another way, grids made up of flat 
airfoil blades deflect more than would be expected from the rule. However, for grids 
with greater deflection the approximation is much better, and the rule may be considered 
sufficiently accurate. 


« 


48 


2321 



IV, 


CALCULATION OF THE STACK ON 
THE BASIS OF GRID TESTS 


4.i. Notation 


In addition to the general notation given under 13. (which is used throughout 
the hook), there is employed in this chapter, beginning with Section 4.3, the notation 
given in Chapter II. Some use is made of the notation of Chapter HI hut In a few 
cases the symbols there used are not the same as those of the present chapter. In order 
that no misunderstanding shall arise, there is tabulated below all the notation which 
was not previously given under Sections 1.3 or 2.1. 


E 


Au*, A XT' 


R 


Mechanical energy content 


£ m£ sec 


Deflection coefficients of stationary and 
moving grids as defined by 3*4 (4), the 
relative velocity being used for the moving 
grid. 

Wall loss function 


7 


h 1 


o i f c| 
u- T a^ 


W r 


& 

8 


- ui 


: c/u. 


Coordinates of a rectangular 
coordinate system as shown 
in Figure 9. 

Unit vectors in the coordi¬ 
nate system x, y 

Local absolute velocity 
£m sec 

Local relative velocity 
£m sec -1 ] 


Half the width of the 
disturbed region in the 
x-direction [m3 


A short horizontal lines over a symbol (i.e., an 
"overlined" symbol) indicates that the corresponding 
quantity is referred to the conditions existing 
after equalization of the flow. 


In 

Section 

4.2 

only 


k" 


5 


_ i __ a 

cr, cr 


Ideal clearance flow coeffi¬ 
cients for stationary grid, 
moving grid, 

Number of blades in stationary 
grid, moving grid. 

Coefficient in wall loss formula 

Radial clearance of stationary 
grid, moving grid [m3 


Where notation is used -which is given neither here nor under 1.3 or 2.1, or 
where the symbols have a different meaning, an explanation is given in the text. 

4.2. General Discussion of the Problem 

This section considers the question: what are the principal errors introduced in 
calculating the energy conversion and the efficiency of a turbamachine stage on the basis 
of test data obtained with a stationary blade grid? It is assumed that this calculation 
is of such a nature that the energy conversion and the efficiency of a single stage 
element are first determined, and then by integration mean values are obtained which are 
characteristic of the stage. However, this technically correct method of calculation 
still involves a rather considerable idealisation in view of the actual processes oc¬ 
curring in the stage. The actual relations depart appreciably from the ideal relations 
in the following respects. 

(a) The assumption of a frictionless fluid in considering a stage element in it¬ 
self involves a considerable approximation, since in a single passage the flow is actually 
three-dimensional and does not proceed along surfaces of revolution 

(b) Circulation around the blade is in general a function of the radius. This 
functional relationship is influenced by the type of flow mentioned under (a), and there¬ 
fore departs somewhat from that which would be expected from the usual theoretical con¬ 
siderations. In particular, under certain circumstances the circulation along the blade 
is not constant in cases -where such constancy would be expected. 

(c) If the axial distance between the individual rows of blades is relatively small - 
as it practically always is in the case of multistage machines - there is a time variation 
in the flow phenomena (the flow being taken relative to the blade row in question). 

This is true even for a frictionless fluid, in which case the flow leaving the grid is 
practically completely equalized at a distance which is of the same order of magnitude 
as the grid pitch. In the case of fluids with friction the effect is greatly magnified 
in consequence of the eddies leaving the exit edges of the blades (the "wake"). This 
periodic change of flow, and the corresponding change in the circulation around the indi¬ 
vidual blades, is related to the formation of eddies which leave the exit edges of the 
blades and result in additional losses. A theoretical Investigation of these phenomena 
may be found in C. Keller, Axial Campx*essors, Page l6j ff. 

it 

(d) The boundary layers on the moving blades are subjected to a field of centri¬ 
fugal force, so that they are not the same as they would be in static tests. Particularly 
large departures from the static relations are to be expected in the neighborhood of the 
blade root, where the boundary layer fluid is accelerated outward, and at the blade tip, 
where the boundary layer fluid accumulates. 

(e) The efficiency of the individual stage elements is different. Energy transfer 
between the elements occurs in an irreversible way, and therefore losses result. 

(f) Wall effects (for example, secondary eddies) may.be perceptible at a consider¬ 
able distance from the walls, particularly in the case of small pitch and relatively short 
blades. 


Further Remarks on (a) and (b) . In order to understand clearly the phenomenon 
mentioned in paragraph (a), consider stages in which all the fluid flows along cylindrical 
surfaces, so that CL stage consists entirely of cylindrical stage elements. If all 
the stage elements are developed on the same flat surface, the blade sections will form 
plane grids. Let the (plane) potential network for all these grids be found by any method, 
and let the grids be so related to one another that if the stage elements are again rolled 
up into their cylindrical forms, the vortex flow law (rc u = constant) will hold at 

the entrance and exit of each blade row. 


50 


2321 




By reason of this rolling up and interlocking of the various grids, the potential 
flow pattern which was absolutely correct for plane flow is transformed into a three- 
dimensional flow pattern which cannot he expected in general to correspond to potential 
flow. The flow patterns of the individual grids were determined independently of one 
another, and are now related only by the condition that rc u s constant at the 

entrance and exit of each grid. The flow is therefore not along co-axial cylinders, 
because this corresponds to no possible condition of equilibrium, but is rather a very 
complicated flow in three-dimensional space. Since the individual blade sections operate 
at conditions other than those assumed in the theory of plane grid flow, their deflect¬ 
ion properties also may differ from those of plane flow, so that constant circulation 
along the blades can no longer be assumed. In spite of this fact, constant circulation 
is in itself attainable, but it will not necessarily be of the same magnitude as would 
be expected from the plane flow pattern of the stage element. 

The analysis here made for the normal stage may be carried over into the general 
case, and will confirm the correctness of the statements made in (a) and (b). 

Remarks on (o) . Calculation of the Entrance Energy For Grids with Non-Uniform 
Velocity of Approach. A periodic 'change in the velocity of approach to a blade grid has 
certain effects in addition to the creation of eddies by reason of the variation of 
circulation. It is particularly to be noted that the kinetic energy is not the same as 
in the case of uniform entrance velocity. It is easily seen, for example, that a particle 
of fluid in the disturbed region following the exit edge of a stationary blade, in spite 
of its lesser absolute velocity, may have a greater velocity relative to the moving blade 
than the particles of the undisturbed flow. These relations, which are the primary object 
of investigation here, are treated by considering a cylindrical stage element developed on 
a flat surface. The problem is to determine the kinetic energy at the entrance to the 
moving blade grid. It is not necessary to treat the stationary blades separately, since 
if the rotor is considered to be stationary and the housing to be rotating, the same 
analysis holds. 

Figure 9 shows the stage element and the stationary coordinate system x,y which 
is used. In order to simplify the equations, subscript 1, previously used for all 
quantities at the rotor grid inlet, is now omitted. Construct on the x-axis a line AS 
of length equal to the stationary blade pitch t*. This line may be laid off anywhere 
on the axis, as the origin of the coordinate system in the x-direction may be shifted at 
will, and the special configuration shown in Figure 9 has no particular significance. 

The distance Ay of the moving grid entrance plane g^ from the x-axis is so s m all that 
the disturbance created by the exit edges of the stationary blades (the zone of decreasing 
velocity) cannot be appreciably smoothed out in the space between the x-axis end g£. The 
velocities at points on the x-axis may therefore be taken as the moving blade inlet 
velocities. 

On the other hand, it is assumed that the entrance ends of the rotor blades, pass¬ 
ing along at a distance A y from the x-axis, have a negligible effect on the flow 
pattern at the x-axis. The effect of the blades at a distance A y is therefore vanish¬ 
ingly small. This simplification, which in itself is quite radical and seems at first 
sight to contradict what was previously said about Ay, is in the present case of small 
importance, since only a general theoretical survey is being made, and a numerically ac¬ 
curate prediction of the flow processes is not attempted. Furthermore, at a short distance 
preceding an obstruction its effect is not very noticeable, but in its wake a much longer 
flow path is required for equalization of the velocities if the fluid is not frictionless. 

The x-component of the momentum passing through AB per unit time is 



'A 


51 


2321 







■where c a and c u are the axial and the tangential components of the local velocity 

c. From 3.2 (9)* this integral depends on y only if the shearing stress integral is not 
equal to zero. The shearing stress integral is here assumed to "be zero, or at least to 
he negligibly small , so that in this case the integral is independent of y. This means 
that the distance "between moving and stationary "blades can "be changed at will, and the 
x-axis correspondingly shifted, without changing the x-camponent of the momentum flow 
through AB. 

The same analysis is valid for the region on the downstream side of the moving 
blade. Since the difference of the x-components of the flow of momentum into and out of 
the moving grid gives immediately the tangential component of the force on the blades, the 
following conclusion can be drawn. 


For non-uniform grid entrance velocity, the integrated mean value of the tangential 

component of the force (and consequently of the work done) is the same as if the non- 

uniform veloolty were permitted to equalize . This assumes that the•shearing stress 

integral is equal to zero, that the deflection properties of the grid are not changed by 
the non-uniform flow, and that the total through flow is the same in both cases. 


is 


The total energy flowing through AB per unit time, referred to the relative motion, 




c a (x) 


w 2 (x) 

-- *f 

2 


P(x) 


dx (1) 


The relative velocity is 


w(x) = c(x) - ui. 


( 2 ) 


so that equation (l) can be integrated immediately for any given variation of the 
functions c(x) and P(x). If the value so obtained is compared with that of the expression 



' _ 2 
w 

+ P 

2 


r 


dx r t*'c a 

w 2 f p 

2 J 


A 


where c a , w, P are values which correspond to complete equalization of the flow, it will 
in general be found that 

^AB -l ^AB 
dt r at 

The total energy flow per unit time through AB (the length of which is equal to the 

stationary blade pitch) is not the same for unequalized flow as for equalized flow. 

The following numerical example will clarify these relations. It is assumed that 
the absolute velocity c has everywhere along the x-axis the same direction, and that it 
makes with this axis the angle = 26°34«, so that tan qC * 0.5. Without limitation 

of generality the stationary blade pitch t*‘ may be assumed equal to 2^ . Instead of 

making the numerical calculation with the velocities themselves, it is more convenient to 
use the velocity coefficients referred to u. The special assumption made as to the 
variation of velocity along the x-axis is as follows: 


52 


2321 




















\ 


5, 


f(x) 


f(x) 1+1 f(x)^, 

2 

f 1.091 - 1.091 / 1 -f- cos x \ for - ^ 4 x < +j 

* 3 / 

11.091 for -'TT< x*c-6 <*nd + £ < X < IT 


where 5 = 21! • T ^ ie va l' UB 1*091 is selected In order to make the mean value of f (x) 

3 

equal to 1 for the interval from - IT to + '7T • Figure 10 shows a plot of the 

function. The dip in the curve Indicates the disturbed region on the downstream side 
of the stationary blades. The pressure is constant along the x-axis, and since the zero 
point of the pressure energy may be selected at will, P(x) is made equal to zero. Since 

^w = the integral of Equation (l) may now be evaluated. 


^AB 

dt 


0.4555 a 5 


If this is now divided by t*’ = 2 Tf , the rate at which energy flows through the 


section per unit length of the x-axis is obtained. 


A further division by 



^a yM 


= 0.5 u reduces this energy flow to the value per unit mass. Finally, the result 

may be divided by u 2 /2 (the "kinetic energy of the tangential velocity") so that a 
dimensionless number, denoted by E', is obtained, which may be compared directly with 
the pressure coefficient -\|/ . In the present case 


E* = 0.290 

The velocity c and the pressure energy P (values existing at a considerable distance 
from the stationary grid) may be calculated from the relations given in Section 3*2. 

Then, from (3), dE^g/dt - 0.440, and after a further reduction as before, 

E* s 0.280 

Finally, it is possible to calculate for purposes of comparison what the energy 
of the relative motion would be for frictionless flow. Assuming in this case that the 
function f(x) has a constant value 1.091 along the x-axis, it is found that 


I# = 0.341 

The subscript 0 Indicating frictionless flow. If this value is taken as 100$, the 
total energy of the relative motion at the entrance to the moving blade grid for the 
three cases may be expressed as follows. 

For frictionless flow 100$ 

For flow with friction, nan-equalized 85$ 

For flow with friction, equalized 82.1$ 

If it is now assumed that the stage element In question has a degree of reaction 
- o.5, then, since X g » 0 frj the pressure coefficient \jr is of the 

order of magnitude 2. The error which enters when equalized flow is assumed in place 
of the actual unequalized flow is 0.01, according to the above calculation. Since in 


53 


2321 







the case of multistage machines this assumption is made twice per stage? namely, for 
the stationary "blades and again for the moving "blades, the total error in the estimate 
of the entrance energy into the "blade grid is about 1$ of the energy conversion In the 
stage. Stage elements with degrees of reaction other than 0.5 will have greater relative 
entrance velocities, and under certain conditions the error may "be greater. 

It is easily seen that such an error In calculation of the entrance energy of 
the Individual rows of "blades can hardly give rise to any appreciable error in the calcu¬ 
lation of the energy conversion in the stage element. To compensate for the greater 
energy which is present in the case of non-equallzed entrance velocity, there is ad¬ 
ditional irreversible interchange of momentum In the blade passages which results in 
further loss. The variation of velocity assumed In the numerical example corresponds 
to an inefficient stationary grid, and this variation was purposely selected in order 
to bring out clearly the differences. In the case of efficient grids the variations are 
much smaller. On the other hand, C. Keller has shown that the losses which arise from 
the eddies due to variations of circulation may become quite large. This is a dis¬ 
advantage of non-equallzed flow which is not taken into account in the preceding analysis. 

The conclusion previously reached (on the basis of momentum considerations) that 
the mean value of the tangential force, and therefore of the work done, is the same for 
both equalized and non-equallzed entrance velocity, may seem to contradict what has been 
said about the change in energy relations. It is necessary, however, to take careful 
note .of the conditions under which the first conclusion is valid, especially the condition 
of constant flow. For a given stage pressure change, a variation of kinetic entrance 
energy results in a variation of flow, or for a given flow It results in a different pres¬ 
sure change. 

Conclusions: It is clear from what has been said in this section that an exact 

prediction of efficiency based on data from a model grid is not possible. This is true 
even if consideration is limited to the basic efficiency, with no account taken of the 
wall losses even with this simplification, the processes in the "uniform 11 middle part of 
the stream are still so complicated that the division into stage elements, the calculation 
of the efficiency of these elements as if they consisted of plane grids with flow non¬ 
varying in time, and finally.the integration along the radius, all require a very marked 
idealization. If the equations given in this chapter are used for the calculation of 
efficiency based on grid tests, therefore, it must be borne in mind that efficiency can¬ 
not be accurately predicted in this way. The value of the method is rather to be found 
in its use to determine the comparative effects of different variables, or the order of 
magnitude of the effect of a single variable. 

It should be emphasized that it is not desirable to refine and complicate the ef¬ 
ficiency calculation too much, since in So doing only a false accuracy would be obtained. 
It is therefore no great limitation that in this chapter consideration is restricted to 
cylindrical, and in particular to normal, stage elements for which incompressible fluid 
relations are assumed. Equations 2.6 (8)-(l7) give the relations between any stage 
elements and the cylindrical elements which may be substituted for them. 

The clearly defined character of this substitution permits a single uniform treat¬ 
ment of turbomachine stages of different kinds which takes advantage of the simple 
relations pertaining to an incompressible fluid. Only in the case of normal stage ele¬ 
ments, for example, is kinematic degree of reaction defined, for the reason that it is 
only in the case of incompressible fluid and of plane flow through a flat grid that the 
well-known simple expression for the theoretical direction of the grid force is applicable, 
and that the geometry of the velocity triangles is so simple that the introduction of f* 
has a practical value (arising from the simplicity and the clarity of the relations which 
may be expressed by the use of p ), 

In cases where the compressibility produces a further effect (that is, not only 
an appreciable average change of density, but, for example, shock waves leaving the ends 
of the blades) it is merely necessary to use in the calculation different grid properties. 
The fact that Mach Number is never brought explicitly into the turbo-machine theory is 
thus Justified. 


5 ^ 


2321 








4.3 Validity of the Equations of Chapter II 

The relations in Chapter II were "based on the assumption of frictionlese flow, 
while in this chapter flow with friction is considered. In order to avoid confusion, 
this section indicates which of the equations of Chapter II remain valid, and which are 
no longer valid, for flow with friction. 

The flow angle and the velocities (or the velocity coefficients) which occur in 
these equations are quantities which are considered to characterize the actual flow (not 
quantities obtained only with frictionless flow). Thus all those equations which express 
a geometrical (kinematical) relation, and also those which are based upon the momentum 
law (or moment of momentum), remain completely valid. Only those equations based upon 
the energy law lose their validity. 

The following change oi definition is also made. In the derivation of the 
equations of Chapter II, the analysis was based primarily on the turbine, but the 
equations could be applied to the compressor by using the proper corresponding values 
or by changing the signs of the characteristic stage quantities when necessary. This 
is the most useful method of presentation to show the general relations connecting all 
types of blading. For practical use, however, it would be rather foolish to make the 
work coefficients and the pressure coefficients of compressors always negative, or to 
work with negative changes of enthalpy. For practical calculations, therefore, A, Iff 

and Kp will be considered positive, and likewise the change of enthalpy will be 

considered positive, so that it will be necessary to change the signs of these quanti¬ 
ties in all the equations of Chapter II when calculating compressors. 

One further observation is to be made concerning the equations of 2.7, which show 

how the various characteristic quantities of the stage element depend on the radius (in 

the same stage) if the law rc u = constant holds at the exit of each row of blades. 

If this law of constant velocity moment is, in fact, satisfied, the equations given 
under 2.7 are valid. In the case of flow with friction, this is not exactly the case, 
and what the law actually is cannot be theoretically determined. 

It would, of -course, be possible to assume equal pressure change in all stage 
elements of the same stage, and to calculate the velocities on the basis of known grid 
efficiencies (for the individual stage elements). However, such a calculation would 
require so many approximations that it would not be Justifiable. Near the middle of 

the stream, the departure from the law c u r - constant is so small for properly 

designed blading that it has scarcely any significance; furthermore. Just because of 
this small departure, an apparent accuracy of calculation would result which is actually 
lot realizable. In the vicinity of the walls, where the departure from the law is very 
large, a mathematical treatment is even more uncertain. 

In accordance with the premise that the calculations are not to be refined to any 
greater degree than is consistent with the general level of accuracy attainable, the 
relation c u r - constant is also used for flow with friction (assuming, of course, that 

the type of blading is suitable), so that the equations of 2.7 retain their validity. 

The question as to what does happen if a blading is not "correctly" designed - that is, 
if it is not designed on the basis of c u r = constant - will be considered in Chapter VI. 

The following table shows which of the equations of Chapter II are valid for flow 
with friction (the asterisk indicating that a change of sign is necessary for compressors). 

Valid Invalid 

2.2 (1)*, (3), (4)*, (8)* 2.2 (2), (5), (6), (7) 

beginning with ^ 


55 


2321 




Valid 


Invalid 


2.3 (1), (7)? 

2.4 All equations, with 
(12)*, (13)*, (14)* 

2.6 (1) to (14), (16) 

2.7 All equations 


2.3 (2), (3), (4), (5) 


2.6 (15), (17) 


4.4, Matching of Two Grids in a Stage Element 


The relations to he developed in this section will make possible the analytical 
transition from the blade grid to the stage element. Suppose that two grids are given, 
one representing the stationary blades, the other the moving blades. The deflection 
properties of the two grids are given by 


AU' = (1) 

A u" = g (A*) (2) 

The problem is that of matching these two grids'in the stage element. Naturally this 
can be done in more than one way. 

For the purpose of the analysis here given, the angles and /^ao are to be 
measured as shown in Figures 4 and 11; otherwise the signs, of the trigonometric functions 
must be correspondingly changed. The double signs in the equations enter by reason of 
the fact that AU 1 , A V" are always considered to be positive. For stationary blade 
(stator) quantities, the upper sign is used if the deflection is in the direction of the 
tangential velocity, and the lower sign if the deflection is in the opposite direction 
For moving blade (rotor) quantities, the reverse is true. 

(a) General Cylindrical Stage Element . Figure 11 shows the velocity traingle for 
the general (non-hamogeneous) stage element with incompressible fluid. Noting that 
T ]_ “To 01113 A U’ are identical if = 1 (and analogously for T 2 

X-j_), it is evident that in general 


T i ■Vi ' 1 " 1 (3) 

-V±AU» (4) 

It follows from the geometry of the velocity triangle that 

- <p ctg ; L — T & ~(r l - I) = f 

These two equations are solved for and the ©xpnossions so 

obtained are substituted in (3) and (4). Thus the following solution is obtained. Let 

A' s 4(ctg oLj, ctg^^ ) - 2 cos ± & u» (5) 


56 


2321 









\ 


B* . 

2 cos 1 A U' 

(6) 

C' = 

-2B* 

(7) 

A” = 

2 cos /3^ t. A U 1 ’ 

(8) 

B" = 

Mctg 0( a o + ct 8/^ 0 ) sin /^ 0 o “ 2 cos / 3 <j o dt A U" 

(9) 

C" = 

2(+ Au" - 2 ctg* rf Bilyk) 

(10) 

Then 

A* X 0 4- B* t 2 + C* = 0 

(11) 


A” X 0 B" X 2 + c" = 0 

(12) 


Furthermore, 


% 2+ 2 

£(ctg<* c o+ ctg^) 

^g ^ + ctg /?<* 


ctg (A^ -T„ 


03) 

W 


Since A U' and A U 1 ' are assumed to be known functions of oC^ and 

Aoo respectively, A’, B*, C’, A", B", C" are also known functions of oL^ and /&<*> . 
If a pair of values q(^ j is selected, the values of the coefficients may be substi¬ 

tuted in (ll) and (12), and X 0 and X ^ may then be calculated, p and X ^ 

follow from (13) and (l4), and thus all values are determined. It is evident that the 
ensemble of matching possibilities of two grids in a cylindrical stage element is in 
general doubly-parametric. 


(b) Normal Stage Element . In the case of a normal stage element there is the 
additional simplifying condition Xq - X 2* If this condition ifi satisfied, it 
follows immediately from (3) and (4) that 


A u« - A u» 

sin**^ sin Ao 


(15) 


The left-hand side of the equation depends only on <^ao ; the right-hand side 
only on /2o^* Equation (15) therefore shows the relation between simultaneous values 

of and (3^ . In a normal stage element of a given grid only one of these two 

angles can be independently selected. Everything else is thereby determined; the matching 
possibilities of two grids here constitute a singly-parametric ensemble. The determination 
of the corresponding values of and is best accomplished graphically, as shown 

in Figure 12. A U’/sinis plotted as a function of , and A If’/sin^*, 

as a function of . If a line such as ABCD is drawn from any as a starting 

point, the end point D gives the value of ^ <*> corresponding to the selected value of . 

For a pair of values c/^, so determined. Equations 2.4 (4) and (6) give 
immediately the values of D and <P ; explicitly written, these are 


57 


2321 








p 


s 


(16) 



c-fcg ft. 


«o 


Ctg oCp *f- ctg^3 


oO 


f- 


Furthermore 


AU r 


K, 


Ctg ^ -f Ctg 


(17) 




Vf-+7‘ 




(18) 


/ K r 




t 2f» ~/°-Xz) 


^Coo \fif l +(\- 

tz =(!-/>) 7 M'/f* <-(>-/>)*' 


(19) 


If the values of £> and if corresponding to the selected pair of angles have 
been found from (l6) and (17), X 2 follows immediately from (l8) or (19), and thus 
the "standard characteristic quantities" of the normal stage element are determined. 

4.5. Efficiency of the Stage Element 
As a Function of the Grid Energy 
Loss Coefficients 

xhe total adiabatic (isentropic) change of enthalpy in a istage may be expressed as 


n ad = ^ad 4- h ad (l) 

This equation is definitely not correct from a thermodynamic standpoint as e<?on 
as friction is included. Because of the loss in the stationary blades, the temperature 
at the inlet to the moving blades will be higher than for an isentropic process. It 
follows that .for a given values of h^ and h^ d will be greater than the value 
given by Equation (1), the discrepancy corresponding to the divergence of the p-lines 
in the entropy diagram (Figure 2). With efficient blading, however, this effect is 
very small, amounting to only a few per cent even for machines with a large number of 
stage- 8 . Equation (l) may therefore be used without misgiving, particularly since there 
is a possibility of correcting the error later (Chapter V). 

The efficiency ^ e of the stage element is defined as follows (cf. Figure 2). 


= & V* - (a) (3) 


2321 


58 


















In these, as veil as In the following equations, the subscript T stands for 
"turbine" and the subscript K for "compressor". 

Pram 1.4 (4) and (5) h e = A A u 2 

2g 

had = Ip A u2 

2g 


Therefore 

TeT - — = ——— = _A_ 

■v HI ^(AP'+dP") 

7eK= X - X 


w 

(5) 


Equation (l) has been used here with A P t Ap' t A P", Returning now - 

to Equation 3*5 (l)> the P Q occurring in that equation is for the present stationary 
blade grid equal to A P'; and for the moving blade grid it is equal to A P". 
Solving 3.5 (1) for P 0 , 



where c n and C 2 have the same meaning as in Chapter III (velocities at a great 
distance before and after the grid) and not the meanings as used otherwise in this 
chapter. In place of c 0 and eg there will now be substituted c 0 and Cj_ for the 
stationary grid, and and Wg for the moving grid. The following equations are then 
obtained 



( 7 ) 


The double signs take account of the fact that & P is always considered 
positive if the change of pressure is in the same direction as the overall, or net, change 
of pressure in the stage; otherwise, negative. In these equations the self-explanatory 
geometric relations ^ 2 = ^ 2 +-(^ 2 , and 2 ■ ( V - l) 2 lf> 2 

have been used. Noting that for the cylindrical stage element Equation 2.2 (4)i quite 
generally takes the form |^| = 2 (-T ' 2 )f and introducing (6) and (7)>l 

Equations (4) and (5) may be written 


59 


2321 













(8) 



T 




(9) 




Both equations are so expressed that numerator and denominator vlll be positive. 
They are valid for the general cylindrical stage element. 


- For the normal stage element it is merely necessary to replace tg ^ T 2 - 
in (8) and (9) . If the normal stage element is one which has been substituted for a 
homogeneous element, as is ordinarily the case, the following is to be noted. According 
to 2.6, the pressure coefficient of a normal stage element which replace a given ele¬ 
ment is defined as the pressure coefficient of the given element. This is not the same 
as that of the normal element considered by itself* since Cq and eg are different for 
the given element; Cq = mQCp. There is a corresponding additional change of pressure 
energy, 

Ap* = ± 1 (c 2 2 -c 0 2 ) = -f c ? 2 (l-m^ 2 ) = 

2 2 u 

- < ' r 2 ^ -t- f2 2 ) (I " 2 ) (10) 

which does not appear for a normal stage element taken by itself. If A P* and Ap" 
refer to the normal stage element, the actual A P is 


A P : dP' t AP" i 

AP*- would therefore have to be added in Equations (4) and (5). If this is done, and 
if at the same time Xq iB m ^ e to 'C'p, (8) and (9) became 





[k.-ri+f*] 


£(r z -x,) 


fe-i)4 ^ 

I-Se 


( 12 ) 


In these equations all quantities refer to the normal element which replaces the 
actual element; only Eq is referred to the actual element. 

■Equations (8) and (9) are more general than (ll) and (12) in that they may be 
used for non-hamogeneous stages after substitution of corresponding cylindrical stages. 
However, it should be noted that the change of axial velocity, and the associated change 
of pressure, have not yet been considered. If it is desired to take account of these, 
it is merely necessary to replace the term ( < Tq 2 - + i^ 2 ) by ( 'Cq 2 _j_ mq 2 (j) 2 ). 

For the normal element, ^ might be expressed in terms of P and 'TP 2 in 

Equations (ll) and (12), in order to express ~X7 e as a function of the standard 


60 


2321 
















characteristic quantities p , , and 'C' 2 • Such explicit expression, how¬ 

ever, would make the equations, already complicated, still more difficult to visualize* 

It has already "been mentioned that the rather complicated form of the resultant 
equations is a disadvantage of the definition of ^ although physically it is the 

most fundamental and the most general method of defining loss. Therefore it is preferable 
to use the equations of this section in complicated cases where the equations of the two 
following sections would not bring out the physical relations in such clear relief. 

4.6. Efficiency of the Stage Element as a 
Function of the Drag-Lift (or Glide) 

Ratio 


Consideration is here limited to the normal element; for others, the equations 
of Section,4.5 are used, with ' and ^ being calculated from £ 1 and £ " 

according to 3.7 (3). Equation 3.^ (4) written for the stationary and moving blades, 
respectively, becomes 

2 1 

AP' = ± ^ -77 (cos0(00 ie'sLnoin) (1) 

^ L 

(co S/ e M te's\n/i M ) ( 2 ) 

b f ' 

The positive sign before ^ is used when the pressure change A P is in 

the same direction as the average change of pressure in the stage; the negative sign 
is used when it is in the opposite direction. The upper sign preceding £ refers to 
an accelerating grid, the negative sign to a diffusing grid. Since - Itk-r.l 

for a normal stage, A U* c = U" w also. If A U* and AU" are 
expressed by 3*6 (5) 

?£'ctg <*.„) = 

2- "fc a- "t 

siL-M sM W4.(| .e'ctsAo) (5) 

€ l \ + € ’ctgoCeo) 

Furthermore 


A = 2Kp = 2 K w ^ At/ =2K Wo0 






W 


The expressions given by (l), (2) and (4) are now substituted in 4.5 (4) and (5) 
for X > A PS and Ap". It is to be noted that ( A P’-J-AP") is divided by 
p 2 in both 4.5 (4) and (5), so that Kb^and appear in the expressions for 

A P ’ and A P" instead of c^x> and In the equation so obtained the value 

of ( ^‘s’/t**) is then substituted from (3), so that ( ^"s"/t*") enters as a 

common factor in numerator and denominator and may be cancelled. For the turbine the 
following expression is obtained. 


61 


2321 





(\-e ctg/?..) 

ctg^) tScji (cos^-*-£sino( 0 o)+ —.%■? ( cos/5^ + £ "sirt/9^) 
Kc^i-c'ctg*..) s * 


This equation, as well as the corresponding equation for the compressor, may now 
he simplified hy expressing E^, , , and the trigonometric functions of cX^ 

and in terras of p and p hy means of Equations 2.4 (3)-(6). This gives 




X 



I- 


-/> + <f£' 
\-P p > 

~r c 


t 


/>*•<*>£* \ 

i-fW 




i + 




<r' 


j 


(5) 


(6) 


The terms which are written in brackets in these two equations are not obtained 
directly in the course of the above derivation, which would lead merely to corresponding 
factors of unity. They are correction factors introduced because the normal element con¬ 
sidered has in general been substituted for some general homogeneous stage element with 
a total pressure energy change differing by A P* from that of the normal stage 
element. Considering now the definition equation of e as given for the turbine by 

4.5 (4), it is evident that ( A p* +• A P" +- AP*j must appear in the denominator 
instead of-only ( A P* -f- Ap"), Alterantively, the following correction factor 
might be used in the denominator. 

AP'-t aP% aP 

Ap'+AP" ' AP -AP* 






If the reciprocal of this correction factor is used in the numerator, the ex¬ 
pression in the bracket is obtained, except that appears instead of X . Since, 

however, X = ^ Q ^ ,and p Q is approximately unity, it is permissible 

to simplify the relations by substituting X for iff ; the correction itself is of 

such a small order of magnitude that it can nearly always be neglected. For compressors 
the situation is entirely analogovis; in the formulas \ may be replaced by 

- 4(1 - p - X 2 >. 


62 


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Aside from the correction factor (which in most cases can he considered to he unity), 
Equations ( 5 ) and (6) of this section are much simpler and more easily visualized than 
4.5 (11) and (12), particularly if p is explicity introduced into tjhe latter equations. 

It may he considered that the expressions inside the parentheses of ( 5 ) and (6) are each 
composed of a "stationary "blade" term and a "moving "blade" term, the former reducing to 

(l - P ), and the latter to p , for frictionless flow. ThiB is a very useful con¬ 
cept in the consideration of individual effects. 

4.7. Efficiency of the Stage Element as a 
Function of the Friction Coefficients 


The derivation of the efficiency formulas is analogous to that of the previous 
section, the assumption again being made that a normal stage element replaces a given 
homogenous element. For the turbine. 






US 

2 


fai «£.+*« K 4 -) 


A = zK r - zk^au"- 

Sin /^oo 


-- AV It k , 

5 in (A^o 


•oO 




K 


Wpo 


If these equations are substituted in 4.5 (4), and if the trigonometric functions of 
and are again expressed in terms of p and by use of 2.4 (3)-(6), the desired 

efficiency formula is obtained. The derivation for the compressor is similar. In 
order that the formulas may be used for any given homogenous stage elements, the same 
correction factors are introduced as in the previous section. • . 


%t = 

7ek = 


[< 


( 1 -^ 0 ) ( 

A 


(1 -rr>J-)(if l +vi) 

A 


1 + ■&■") 


4.8. Basic Stage Efficiency 


( 1 ) 


( 2 ) 


The basic stage efficiency q is defined as that efficiency which would be 

obtained if there were no wall and no clearance losses. It is apparent frcrn the relations 
given under 4.2 that -p q may be calculated from 'V Q by integration along the radius. 
It may be shewn that ' 



63 


2321 












since the flow through a stage element is dG - 2 7T r 2 <lr 2 ^ ) 2 U 2 = 

2 TT LO y 5 y 2 2 (^2^2. The factor 2 th© integration sign is included 

only for the sake of formal exactness. In practice V at the rotor exit can always he 
considered as a constant, so that £ 2 might "be omitted. 


If the axial velocity at the moving blade exit is constant aiong the radius, then 
- constant, so that the equation, with Y 2 omitted, reduces to 





7 2 7 e dy S 


( 2 ) 


This form is particularly useful for normal stages, where the subscript 2 becomes super¬ 
fluous. In the following equations of this section the subscript will be omitted, since 
it will be assumed that all stage elements are cylindrical. 

The integration itself can be carried out in the usual way, the value of the 
integrand being calculated for various values of y, and the integral itself then being 
evaluated with the help of a planimeter. In all practical cases it is sufficient to 
determine Q for only three values of y, and to assume that the function^ e (y) is a 

quadratic parabola passing through these three points. The numerical calculation is facili¬ 
tated by working with the loss coefficient e « 1 - ^ e rather than ^ e , since 
the per cent change of the former at different radii is much greater. Hence 

^0 = 1 - 7° 1 ^"7- £ y i>' e ( 5 ) 

How let 

£ e = a 4 - by +• cy 2 


The coefficients a, b, c are to be determined so that at three specified radii, on three 
radius ratios denoted by y 2 , and y ? , ^ assumes the values ■<£ Ql , e2 > 

and respectively (note that the subscripts 1 and 2 do not have the cane 

meaning here as elsewhere). The following system of simultaneous equations is then set 
up. 


a 


'° y i 

+~ 

a 

-h 

"b y 2 

■h 

a 

+ 

by 3 

+■ 


Tills system may be solved for a, b, and c 


Cy l 

- 

^ el 

2 

cy 2 

= 

•<ea 

2 

cy 3 

= 

^ e3 


conveniently by the use of determinants. 


64 


2321 






*&1 v\ 


2 

1 <el Yl 

a - 

%2 V 2 yf 

h 1 

1 <e2 4 

A 

V 3 y§ 

s A 

1 ^5 4, 


c - 


1 7 1 tfel 

1 Vo *tl 


1 v 3 % 


e3 


( 5 ) 


where 



After the coefficients a, b, and c have "been determined, the integration may he carried 
out. In fact, it may he carried out once and for all and expressed in the general form 


i -y 0 




(ay + hy£+0y3)dy = 


(7) 


= a th ”X b 00 t c X c 00 


where 


Tb = g (Y 3 - 1) 
My 2 - i) 

Xc = Y 4 - 1 

2CY 2 " 1) 


(0) 


(9) 


Values of h ^ £ c are tabulated in Table 2. The analytical integration here 

suggested has an advantage over graphical integration in that the calculation need he 
carried out only once in order to determine "7? q as a function of Y from (7). It 

might he thus determined, for example, for the different stages of the same machine. 


In practice it will he necessary in only very few cases to determine ^ 
by integration. In most cases a simple arithmetical average will he sufficient, 


from y e 
or the 


65 


2321 




















a ssumption may "be made that the "basic efficiency of the stage is equal to the efficiency 
'V e of the "mean" stage element (that is, of the stage element at diameter Ifo). 


4.9« Clearance Losses 


Clearance losses can "be determined only "by experiment. The clearance loss formulas 
expressing the results of certain experiments can, indeed, assume any desired form, since 
it is necessary merely to determine coefficients in such a way that the formulas express 
accurately the results of the tests. It is, however, very advantageous to choose the 
form of the formula in such a way that it corresponds to the physical mechanism of the 
process as much as possible. This is not the case for most of the well-known clearance 
loss formulas. In part they are not even dimensionally correct, and therefore do not 
express any fundamental physical reqifLrement which must he satisfied in all cases. It 
is proposed to determine here a suitable form for such a formula; that is, a form which 
has a real physical basis. 


Suppose, for example, that the clearance loss 
by the widely-used formula 


Z\i B p is expressed (in heat units) 


M. 


k cr 

i 


"sp " “ r ad 

where k is an empirical coefficient, the clearance width, and hg^ the adiabatic heat 


drop in the blade grid considered. If in this grid there is a change of direction only, 
and no pressure change, then h^ = 0. This means that the formula fails completely 

in this case, because there is a clearance between the end of the blade and the passage 
wall, and there is unquestionably a loss originating here even if = 0. This 

shows immediately that the formula does not correctly express the actual physical process. 


Consider now a stationary blade grid, and let there be a radial clearance O' 
between the ends of the blades and the rotor, as shown in Figure 13. The effect of the 
clearance is primarily to change the flow pattern in such a way that a part of the fluid 
is not subjected to a change of direction in the stationary blade grid. Let Bq be the 

total moment of momentum of the fluid entering the stationary blades per unit time, and 
let Bj* be that moment of momentum which the fluid leaving the stationary blades would 
possess if the clearance were zero. In consequence of the clearance actually existing, 
only a fraction (l - A ) v of the amount of fluid flowing through the stationary blades 
is actually deflected; and the fraction A flows through without deflection. This 
division into two parts (l - A ) and A is of course not an actual physical phenomenon, 

but the division may always be so assumed as to be equivalent to the actual phenomenon. 

If B^ denotes the flow of moment of momentum at the exit of the stationary blade grid, 
as modified by the effect of the clearance, it is evident that 


Bq r Bq*(l - A ) +- B 0 A 

No account has been taken of the fact that the rotor through the action of viscous 
forces drags along with itself to a certain extent the portion A of the fluid, and 

thereby effects a certain change in the moment of momentum. It will be shown later that 
this approximation is not very great, and the effect of viscosity at the boundary walls 
will be neglected. 

The moment of momentum equation (Euler equation) for the axial space between the 
stationary and moving blades (more accurately, for the annular space between the planes 
E* and E", Figure 13) will now be formulated. If B^’ is the moment of momentum passing 

through the plane E", the conservation law gives for the external moment M acting on 


66 


2321 



the region considered 


M = Bj." - B x 

Since M : 0 if the effect of the vails is neglected, 

V : B 1 : Bj* (i - A ) + B 0 A 

Whatever mixing process may occur in the Bpace between E’ and E", therefore, the moment 
of namentum is the same at the entrance and exit of the moving blade grid. 

It is clear that the effect of the viscosity from the outside boundary wall can 
be neglected, since it has nothing to do with the clearance loss at the inner boundary 
wall (i.e., the rotor surface). Neglect of the viscosity influence at the rotor itself 
is also permissible. Although the moment of momentum of the portion A is changed by 
reason of this viscous effect, with a corresponding change in the turning moment acting 
on the moving row of blades, it is the rotor itself which effects this change, so tliat 
the net turning moment remains constant. Of course it is not intended to imply that it 
is entirely immaterial that the rotor constitutes an inner moving boundary wall. On 
the contrary, this fact has a decisive influence as the boundary layer builds up, but 
the effect is accounted for in A . 

From this analysis the following conclusion may bo drawn. Whatever may be the 
actual process in the boundary layer, it is equivalent to one in which a certain fraction 
of the flow A is given no deflection whatever, so that this portion of the fluid enters 
the mpving blade grid with the same tangential component with which it enters the station¬ 
ary blade grid. Since in multistage machines the velocity of entrance into,the station¬ 
ary blades is almost always approximately eqiial to the absolute exit velocity from the 
moving blades, this is equivalent to saying that the portion A does no work, or has no 
work done upon it, in spite of the fact that it flows through the clearance space only 
in the stationary blade grid, and regardless of what change of enthalpy may occur in the 
stationary blades. 

In an entirely analogous manner, the clearance loss for the moving blade grid may 
be determined, and the same conclusion will be reached; namely, that whatever fluid flows 
through the clearance space at the tips of the blades represents a loss for the entire 
stage - not merely for the moving blades. The introduction of the isentropic heat drop 
of a particular row of blades into the clearance loss formula is therefore theoretically 
wrong. The formula itself is rather obscure in form, and another will now be derived 
which will be clear and expressive in form as well as theoretically sound. 

Consider a row of open-ended blades with tips at a radial distance O' from the 
bounding wall. The geometrical shape of the ends of the blades is assumed to be given. 
Test data can be applied to another case only if geometric similarity exists; that is, 
if the ratio of the clearance width toy some specified reference length of the grid, (as, 
for example, the blade pitch t*) is the same. In the formula to be derived, therefore, 
the empirically determined flow coefficient k must be a function of <J~ /t* for a 
given grid shape. 

The derivation will be based on the principles of the airfoil theory. For the 
actual profile at the blade tip there is substituted a straight line (or thin "flat plate") 
which has a length equal to the chord s (see Figure 14), and which maless an angle OC^ 

with the grid axis, so that its direction is the same as that of c ^ . There is a 

lift acting on this plate, but no drag, so that the resultant force is perpendicular to 
the plate (a simplification which in this connection may be accepted without further 
Justification). The lift force S a is considered to be produced by a pressure difference 

A p uniformly distributed along the entire length s, as indicated in Figure 14. 


67 


2321 


As a result of this pressure difference, there is a certain flow through the clearance 
space at the blade tips, the amount of which may he calculated by the usual formula and 
expressed as a fraction of the total useful flow. The lift force iel 


so-that 



A p 


s 




The clearance flow velocity is 


I 



so that the leakage flow for a single blade is 



The number of blades in the grid, or ring, is '7T D/t*, and the leakage flow for the 
entire ring is therefore * \ 

G sp - D —L. C o®V^ 

t* 

The total flow through the ring is 


o = : si ncX.fl 

where XX is the annular area of the ring. If the flow coefficient k, which is to be 
empirically determined, is now introduced, the formula for the clearance loss becomes 


^ sp = k . - k TTPCT s_ ~^ot (1) 

G t* sin ^?C o0 

Substituting for ^ from 3.6 ( 5 ), with £ : o, 

a , * 

? sp = k 'TTPCT V 2 t* ^ (2) 

XX sin^^ 

This formula may now be written separately for the stationary and the moving blade grids, 
using the proper distinguishing symbols. 



Vlt" ) 


•frV r ' 

n' 


Vfp AU' 

S l n ^oo 


( 3 ) 


68 


2321 



















Tr D5<3~" 

XI' 


-g**Z\LT 


w 


It will be noticed that 


'vrTxr/h. 


is simply the ratio of the clearance area to the 


total annular area. The factor 2 under the square root sign is-omitted "because it can 
he included In k. If geometric similarity exists, s/t* might also he omitted, hut 
the formula is physically more expressive if this ratio is left in, since it may he seen 
that for a given deflection A the clearance loss decreases as s/t* decreases; or 


as the pitch and hlade loading increase. Of course, increase of pitch will also affect 
the value of k (in vhich direction cannot he predicted), hut it is to he expected that 
the influence of the term under the radical will he controlling. The total clearance 
loss of the stage is 




~5 


tt 

sp 


(5) 


In this equation there are no further factors to he introduced to indicate the 
"weight” to he assigned to the individual losses hy reason of the relative adiabatic 
heat drops in the two rows of blades. As appears from the analysis, both losses involve 
the entire energy transformation in the stage. The derived form of the formula shows, how¬ 
ever, through the factors sin OC^ and sin/3^ in the denominators of (3) and (4), 

respectively, how the heat drop occurring in the row of blades which is considered in¬ 
directly affects the magnitude of the clearance loss. 

For a grid with purely directional change and no net change of pressure, s 90° , 
and therefore sin - 1. For an accelerating grid (X^is less than 90°. The 

greater the acceleration, the less the angle, and therefore the greater the pressure 
change. As the angle decreases, sin also becomes smaller, and becomes 

greater. This is also true of diffusing grids, since sin cannot exceed the value 

unity corresponding to a grid which merely deflects. The "weight" to he assigned to the 
loss in each of the rows of blades is therefore already Included in the clearance loss 
formula as given, and is included in a way which is absolutely correct from the physical 
standpoint. 

The proposed formula applies only when the ends of the blades are free; i.e., when 
the blades are of the cantilever type. If hands or shrouds are used, so that there is 
no open space left between the ends, the computed total useful flow may be vised direct¬ 
ly in subsequent calculations with an empirically determined coefficient. In contrast 
to the previous case, the actual total pressure difference across the row of blades is 
determinative In this case. Here also the flow through the clearance space represents 
a loss for the entire stage, and not merely for the particular row of blades in question. 

It is practicable to define k’ and k" in such a way that the clearance loss is 
accounted for, not by subtracting "“^fsp from 0 , but rather by multiplying 'T? Q 

by (1 - ^ ). The latter method is a purely formal convention which has been found 

most convenient in practice. 

4.10. Wall Losses 

At the bounding walls of the meridian passage there are boundary layers which 
change the flow relations completely as compared with those existing in the undisturbed 
middle portion of the stream. In turbines, because of the relatively large deflection 
of the fluid, there is a clearly defined secondary whirl. In compressors the thickness 
of the boundary layer becomes relatively very large because of the pressure increase. 


69 


2321 






The losses connected with these phenomena are called wall losses, and although in a 
purely physical sense they cannot he separated from the clearance losses, because the 
clearance affects the boundary layer, nevertheless this distinction has a practical 
physical significance. 


Assume, foi example, that tests have been carried out on an experimental rachine 
with different clearances, and that the total wall loss (including the clearance loos) 
has been measured in each test. If these measured values are extrapolated to zero 
clearance, the residual loss so determined iB what is here called "wall" loss. It is 
denoted by the symbol R . 


If the total wall loss for any clearance is 



sp> 

since the difference between ^and p is obviously a result of the clearance 

effect, regardless of how the physical mechanism actually operates. Figure 15 shews 
schematically how W ’ De separated into and ^g-p by extrapolation. 

This clears the way also for the determination of the coefficient k of the clearance 
loss formula. 

In theory it is Immaterial whether such measurements are made on an actual maohine 
or on a stationary model. It is to be noted, however, that it is exactly these wall 
effects which are most difficult to investigate in the case of a stationary model, 
particularly since the moving wall opposite the tips of the blades cannot be simulated.- 
In order to obtain useful quantitative results, rather than mere qualitative conclusions, 
measurements must be made directly on an actual machine. This is really true of all 
losses, including those occurring outside the wall region, but the profile properties can 
be determined well enough by stationary model tests so that valid comparisons, at least, 
are possible. Static researches have in themselves the great advantage that the details 
of the flow can be better observed. 




r<t 




R 


The wall loss formula is derived in a technically correct manner as follows. In 
Figure 16a let the stream surface ABC bound the region within which there occur wall 
losses ascribable to the outer wall, and let a, b, and c be the annular areas at B, 
and C, respectively. At b the integrated total energy is smaller by the amount A E’ 
than at a. Furthermore, let the integrated total energy of the relative motion at c be 
smaller by the amount A E" than at b. (For the region between a and b the energy 
corresponding to the absolute velocity is taken.) The outer wall loss of the stage is 
then A Ea « A E» + A E*. 


If toe dimensions of the two grids are changed as indicated in Figure l6b. with 
preservaxicn of geometric similarity, it is clear that the dimensions of the region be¬ 
tween ABC and the wall will also change. Strictly speaking, exact geometric similarity 
does not exist between this new region and the original region, hut the unimportant 
departures from such similarity may be neglected. If it is considered that the change 
of the annular areas a, b, and c is practically to the same scale as the change of blade 
dimensions, it is evident that the wall loss, referred to the total stage flow, must he 
proportional to the absolute dimensions of the two blade grids. This follows from the 
fact that the flow between ABC and the wall is proportional to the absolute dimensions, 
and the loss per unit of this flow remains the same. ' 


If 5 is the thickness of the boundary layer considered 
point), and D s is the outer diameter as indicated in Figure l6, 

approximately equal to 'TT D s <S . Since 5 is proportional 

the rotor tip, the annular area is proportional to 7T D t *" 

* S 8 • 


(at any fixed reference 
the annular area c is 

to the blade pitch at 

The wall loss is 


70 


2321 




proportional to the ratio of this annular area to the total annular area XL , so that 


X = % 

^Rs 3 i 




Ci) 


n." 

Similarly, the wall loss corresponding to the inner houndary wall is 


~-*Rn = % n 


* W 

XL" 


( 2 ) 


These equations may he written in an especially convenient and expressive form 
hy noting that 


D s * YDnJ t|" = Yt*"; ^ ^ (Y^-l) 

n -**- jj- 

If these relations are substituted in (l) and (2), and if t*" is expressed in terms of 

n 

the number of rotor blades by the relation 


the total wall loss 



z“= 

"f KS +- "^Rn 10 given T)y 



4ir 

z 



Y 2 ■ 4 ?n _,1 

Y 2 -1 Y 2 -! 


(3) 


The coefficients ^ and n have an evident significance. Each is the 

thickness of a boundary layer (in which the efficiency is zero) divided by the corre¬ 
sponding blade pitch (t*" or t*"). The number of stationary blades, z', may be used ; 

s n 

instead of z" if the coefficients are correspondingly defined, since with geometric 
similarity the ratio of z' to z" is necessarily fixed. 

For a given type of blading, the expression in the brackets is a function of Y 
only, and the formula may therefore be written in the simple form 




(*) 


where R is a quantity which must be determined empirically, and which for a given 
type of blading may be expressed as a function of Y, the maximum radius ratio, or Of 
l/D^, the ratio of the blade height to the mean diameter. 

It is to be noted in (3) and (k) that z", the number of blades, occurs in the 
denominator. For a given shape of grid (or even for different shapes if these are ex¬ 
pressible as functions of the radius ratio y), the wall losses become less as the number 
of blades increases, or as the blading becomes "finer", and this fact points clearly to 
the proper method of experimentally determining the wall loss. 


71 


2321 









Such a determination requires tests with grids having the same values of Y and the 
same variation of shape with y, "but different numbers of blades z". If the results are 
plotted as a function of l/z" and extrapolated to l/z" = 0, the basic lose is differ¬ 
entiated from the wall loss. Each of the test points which determine the extrapolation 
curve is in itself the result of an extrapolation to zero clearance* so that in thip 
treatment the extrapolation to zero clearance Is still tacitly assumed. 

Just as in the case of the clearance loss, the convention will be adopted tha t 
the effect of the wall lose is to be included in the formulas by means of a multiplying 
factor (1 ■ "Sjj) , and not by subtraction of ^ . 

jR 

4.11. Stage Efficiency, Pressure 
Coefficient, and Work Coef¬ 
ficient of the Stage 

The stage efficiency is 

ts = ?o u - S sp) (1 - r R ) (1 ) 


Formula 2.6 (2) for the work coefficient A g and the pressure coefficient V/q of 

tho stage has already been given. A certain difficulty now enters which will be considered 
fireo Without including the effect of wall losses. Assume that in a given case \ s 

and "7 s are calculated from 2.6 (2), and that ^ Q is calculated from 4.8 (l). 
will it be found, on checking back, that ^0 = /V S the case of a turbine, 
and that _ y g in the case of a compressor? Such a conclusion can- 

n0t *7 drawn directly from the equations, but on the other hand at least one of the three 
quantities y 0 , /\ s > y g would not be defined in a suitable way if this simple 

hold ‘ As a practical mtter > the difficulty is resolved by reason of 
the fact that the meridian flow pattern and the weight flow through the individual stage 
elements (that is, the function f (y 2 )) adjust themselves s^ae to produce tois ^ 


- . Wal1 108803 111 the analysis results in a further canrolication. 

Vfoat has previously been said is, of itself, still valid. However, it is necessary to 

“of ion 2 ? ln ^ex^st th LT zones ' the rr 1 ,!de - 

r ’ A > ana r/ g which exist in these zones would have to 
be used, and the effect of the wall losses would then be included in . in the ^7 

of the stage element would have to be included the effect of interchange' of energy be-' 9 

Stamen+ 8 ln° r ? S ele f }nts re8ultin S turbulence (strictly speaking, this 

statement is of more general application, and does not apply merely to the wall zones} 

convenlent to c °ntinue to consider the wall losses as an additional 

& miforra ba3lc flov - If this is done, there ^e again tSo 
possible concepts which may underlie the further theoretical treatment. ^ 

First Concept: In consequence of the wall effect, there will bo in -^n 

(and therefor, to the entire stage) less TO r* done per mirZTnuH a Sh J and 
more work required per unit mass by a compressor. 7 turbine, and 

Se cond Concept : In consequence of the wall effect, a greater pressure difference 

is:: =ss s : sz =: s s r. star’- • 

s,‘rs.':,s,?,: h :.ssr s s- 


72 


2321 






Included in the value of 


The "basic equations are 



used in the calculations. 



T OK = (2) (3) 

A S 


The isentropic change of enthalpy, h ad , and the change of enthalpy corresponding to 
the effective work, h 0 ("both for the entire stage), are 


h ad = lKs h ( u 2H) (1+) 

h eT = Ts h ad . As^ 1 " ^sp)(l“ ^R)h(u2H) (5) 

» 9K : _i£L_ = _8_ h(^) (6) 

? s a- ^s P )(i- ’S'r) 

\ 

Substituting (l), (2), and (3) in (5) and (6), 

h oT : Ts heK * _2!k_ Hasa) (7) (8) 

-- 

These equations show that it was in fact advantageous to introduce the wall and clearance 
losses into the calculation by means of the multiplying factor (l- '■£’’) rather than by 
way of an additive term. 

The reason for favoring the first of the concepts just mentioned is as follows. 

The work coefficient \ of the individual stage element is made up of purely kinematic 
quantities, and therefore is itself a purely kinematic quantity. If the second concept 
were adopted, the effect .of the wall losses would have to be included in the integrated 
mean value of \ s , which would then characterize a stage, not on the basis of the 
relations existing in the middle of the undisturbed flow, but rather by means of certain 
average relations which include the effect of the wall losses. However, only the kine¬ 
matic quantities of the basic flow are of present interest, and the physical significance 
and clarity are obscured if the effect of the "degenerate" kinematic relations in the 
wall zones is distributed over the entire flow. 

Usually the middle stream surface is chosen as the principal stream surface; that 
is, point A (Figure l) is taken midway between the inner and outer boundaries of the 
meridian passage. In many cases it is then sufficiently accurate to dispense with all 
integration along the radius, and to let q = ^eH, = 'Vk, A s = Ah, 

where the subscript H refers to the principal stream surface. The equation = A H 

could not be written if the second concept had been adopted, and consequently the re¬ 
lations would not have been so clear. 


4.12. Effective Change of Enthalpy. 

Thermodynamic Efficiency 

The actual change of enthalpy in the stage, h, is in general not exactly the 
same as the change of enthalpy, h Q , which corresponds to the effective work done. In 

addition to the change corresponding to the external work, there is also a change 
corresponding to the difference, between the kinetic energy of the fluid at the entrance 
and at the exit of the stage. For the individual stage element 


73 


2321 












h = he f _JL_ (o 2 2 -o 0 2 ) - h e 1 [-T 2 2 + ^ 2 2 

— TT? 0 (T 0 +^0 2 ) J htug) 

in which the upper sign is used for the turbine, the lower for the compressor. For 
homogeneous stage elements, 

h = h e ~ [" ( ^2 2 -f 2 ) J h ( u fi) 

For the complete stage there is an exactly similar relation except that the term written 
in brackets is replaced by the corresponding integrated mean value. 

The integral expression is easily derived, but has a somewhat complicated form, 
and since it always has a very small effect in the cases here considered, it is sufficient 
in practice to use the following approximate formula. 

11 = ^e 2s) “ “oh ( ^ OS *#* Os)J h(u^j) (l) 


which for homogeneous stages assumes the form 


h e~| ( T 2Q i- ) (^“oh) 


( 2 ) 


These formulas are especially accurate if 


^ 23 . - 

The thermodynamic stage efficiency t is defined for turbine and compressor 
by the following equations. 


7 


tr 


*ad 


?* 


n ad 


(5) (4) 


This efficiency differs very little from ^ s . . It is distinguished by the fact that 

it includes only those losses which are losses in the thermodynamic sense; that is, which 
are caused by irreversible processes. It is evident that ^ ^ can be calculated at 

once, since it is merely necessary to omit the correction factors inside the brackets of 
4.6 (5) and (6), or of 4.7 (l) and (2), in order to obtain the thermodynamic efficiency 
of the individual stage element. The value of for the entire stage may be ob¬ 

tained by a procedure similar to that used to obtain 

It is necessary to use h and ^ ^ whenever the actual thermodynamic changes, of 

state in the stage are considered (as, for example, in plotting a process on an enthalpy - 
entropy chart). 



74 


2321 










V. CALCULATION OF THE MULTISTAGE TURBOMACHTNE 


5.1. Notation 

The notation of Sections 1.3 and 4.1 is used In this chapter, with the exception 
that the following simplifications are introduced. 

The subscript S, hitherto added to quantities which characterize an entire stage 
(as distinguished from an individual stage element) is omitted, since individual stage 
elements are not considered at all in this section. For example, y is written instead 
of Ys- The tangential velocity u will always be understood to mean, if nothing 

further is Specified, the linear velocity at the principal diameter, and the subscript H 
previously used to indicate this diameter will be omitted. 

The following special notation is used 


R 

^sp 

% 

L 

N 

Z 

f 

f oo 

1 * 

i 

v 

A 


Flow correction factor (overall) 

Flow correction factor (correcting for clearance effect only, 

Flow correction factor (correcting for other wall effects only) 

Work factor (see 5*3 (13) and (l4)) 

Power ^continental hp or kw] 

Function used in obtaining number of stages (see 5.3 (8) and Table 3) 

Heat recovery factor of turbine, or heating loss factor of compressor 

Basic value of f (for an infinite number of stages), as given in 
Table 4 

Clearance flow ratio (see 5.b (3)) 

Enthalpy [kcal kg -1 J In same equations, where confusion will 
not result, this symbol is also used to indicate the stage number 

Specific volume [v? kg” 1 ] 

Taper ratio (see 5.3 (7)) 


Subscripts: 

i i-th stage 

(1) at entrance to group of stages 

(2) at exit from group of stages 

1 initial condition 


2 final condition 

Subscripts I, II, etc., indicate the particular group of stages considered. Overlined 
symbols (short horizontal line above symbol) indicate quantities which are mean values 
for the group of stages in question. 

The following is to be noted in using the subscripts (l) and (2). If dealing 
with a homogeneous or approximately homogeneous group of stages, subscript (l) indicates 


75 


2321 



the entrance to the first rev of blades actually belonging to the homogeneous group, and 

(2) indicates the exit from the last row of blades actually belonging to the group. This 
must be kept in mind particularly in the case of certain compressor blading (with degree 
of reaction less than 1) for which there is provided a special set of inlet guide vanes, 
considered as distinct from the rest of the stationary blading, and followed directly by 
a row of moving blades which constitutes the first normal row. 

In such a case the inlet guide vanes serve only to impart to the fluid the same 
rotary motion which it would have if it had come from the stator of a "zero stage". The 
last row of blades is then a stationary row, and it was assumed in the equations applying 
to homogeneous blading that this row is the same as the other stationary rows. If this 
is not the case, the last stage may be considered to be a "group* 1 by itself, with its own 
characteristic data. Even more simply, the effect may be included in the efficiency p 
of the diffuser which follows. 

The quantities X and given in the equations might, depending upon the type 
of blading, refer to any one of the locations 0, 1, or 2 3n the individual stage. A , 
more definite reference is given only indirectly in the equations through the use of the 
subscripts (l), (2), whereby for any given type of blading the location may immediately 
be inferred. 

The principal diameter Dg will be understood to be the diameter to the principal 
stream surface, measured at the plane in question. It is suggested that either or 

be used as the principal diameter. With the former, the relations become very simple, 
since the inner boundary of the meridian passage is the principal stream surface. On 
the other hand, if Dm is used, in the general case (see Figure 1) this can be the dia¬ 
meter to the principal stream surface only at a specified place (for example, at the 
exit of the row of moving blades, plane 2, Figure 1). In this chapter, however, the 
exact flaw pattern inside the individual stage is not considered at all, so that this 
disadvantage associated with the use of 1^ is of no real significance. In every plane 

the and X may be referred to 1^, regardless of the fact that this principal 

diameter does not correspond to the same stream surface. The validity of equations 2.6 

(3) and (4) is not affected thereby. 


5.2. Thermodynamic Basis 


In an isentropic change of state with initial conditions 
pressure pg , the work done, including the work of delivery, 
unit mass) 



is (in heat 


I 


, and final 
units per 


( 1 ) 


may be read directly from an entropy chart. It is particularly convenient to do 
this in the case of steam turbines, because Had is usually quite large, and an accurate 
value may be read from the chart. For axial flow compressors and gas turbines, however, 
Had ls as a **ule quite small, so that often a sufficiently accurate value cannot be 
obtained in this way. On the other hand,' the very fact that the change of condition is 
relatively small in the latter case makes a calculation comparatively simple, since the 
medium may be assumed to have the properties of an ideal gas. In such a case 


is. 

H aiT ' / ' Rt [i]^TT 



76 


2321 






Te-J 

1e 


(3) 


H 


adK = * RT [i] t _. 



If the change of state occurs in several successive stages, without losses, then 
^ad - ^ hg^ ^ , where hg^ ^ is the isentropic change of enthalpy in the individual 

stage. However, with losses the sum of the isentropic changes in the individual stages 
is not equal to Hg^ . Figure 17 explains these relations. Since the loss appears as 

heat in the fluid, the enthalpy of the working medium at the exit from, a row of Hades 
or of a stage is always greater than if the process occurred without losses. For this 
reason there is always an increase in the heat available for a succeeding stage in a 
turbine, while a corresponding amount of additional work is required in a compressor. 

This may be expressed mathematically by the relation 

I Vl : (1 + f) Baa (I*) 

where (l •+■ f) is the heat recovery factor of the turbine, or the heating loss factor 
of the compressor. 

Before a more accurate determination of the magnitude of f is attempted, the 
mechanism of the process must first be considered. If any loss occurs during the flow, 
the "lost” energy usually appears first in the form of the kinetic energy of a turbulent 
motion superimposed upon the main flow. This turbulent motion is gradually dissipated 
into completely disordered molecular motion, the kinetic energy of which is, in fact, 
nothing else but heat. Even if one were in a position to observe the process completely 
in all its details, it would not be possible to say how long the "lost" energy is present 
as energy of turbulence, or to specify the exact moment at which it becomes heat, because 
a sharp line of demarcation cannot be drawn. 


The most extreme assumption which can be made is that, as soon as any loss what¬ 
ever occurs, the "lost" energy is transformed immediately into heat, which is distributed 
uniformly throughout the entire working medium. Such an assumption over-estimates the 
effect of the losses, and is thus on the safe side; that is, the calculated value of f 
is too great. If it is further assumed that the overall change of state consists of an 
infinitely large number of infinitely small changes, all occurring with the same effi¬ 
ciency b (so that for a turbine dieff z 7?t di ad and for a compressor 

di eff = ^ad/f t)> then the magnitude of (l -f- f) may easily be determined from 
the First law of Thermodynamics, if the medium is considered to have the properties of 


an ideal gaB. For a turbine 


I 


L, J- 


p ri] 


l?t 


ie~i 

le 


7t 


and for a compressor 



ie-i 


(5) 


l + -Po® - 


no. ^ 

. / \ _L AjzI 


(VO 


jg- 1 


( 6 ) 


The derivation of (5) may be found, for example, in P. Lorain, "Steam Turbines", pages 
219 ff, while Equation (6) may be derived by analogy. The subscript indicates that 

the change of state is considered to be divided into an infinite number of elementary 
changes of state, and that the losses are immediately and completely transformed into 

heat. 


77 


2321 











Ordinarily calculations are not made with f ^ "but with a vaj & of f which takes 

into account the finite number of stages. This assumes that the entire L^bb of a stage 
(or of a row of blades) appears immediately as heat at the exit of the stage (or row), an 
assumption which is evidently somewhat arbitrary. To offset this arbitrariness, it is to 
be noted that the method simplifies the calculation a great deal, and. that the vhlue of f 
at most amounts only to a few per cent, so that a significant error cannot arise, A 
formula for f will now be derived from a simple analysis of the temperature-entropy 
diagram.* 

Figure 18 shows on such a diagram the process of expansion in a multi-stage turbine, 
the area ABODE indicating the work corresponding to an isentropic change of state. AG is 
the ourve which would be followed if the efficiency were the same for each infinitesimal 
change of stage, and if each loss of energy were immediately transformed completely into 
heat (assumptions on the basis of whioh f is calculated). It is apparent from the 
chart that 

1 •+• f - Area AGBODE , 

00 “ Area ABODE 

s Area AGB 
Area ABODE 


The «rea AGB therefore represents the heat recovered. To allow for the fact that the 
number of stages is finite, only the cross-hatched portion of the area AGB is taken, so 
that the triangles which are not cross-hatohed are to be subtracted. These might be 
considered as approximately similar to the triangle AGB, and if it is assumed that the 
individual stages have the same isentropic heat drop, the triangles are approximately 
equal to one another. If z is the number of stages, the area of one of the small 
triangles is AGB/z^ , and since there are z small triangles to be subtracted, 



The same considerations apply without modification to the compressor, and lead to 
the same formula (7). This formula can always be used with sufficient accuracy for 
ordinary calculations, even if the relations depart from those here assumed, since the 
error would be very small. Values of f^ , computed from (5) and (6), are plotted in 
Table 4 as a function of efficiency and pressure ratio for k s 1.3 and k a 1.4 


This analysis also shows how the error introduced into the calculation of the stage 
efficiency by the assumption that had « kJid +- ^ad Eiay corrected, if necessary, 
since in making this assumption'it is exactly the heat recovered in the individual stages 
which is neglected. For this purpose it is necessary only to replace the individual stages 
by the individual rows of blades, and to carry out the process in an analogous manner 6n 
the entropy diagram. The only difference is" that there will be two kinds of triangles 
which are not crose-hatched-one for the stationary blades, and one for the moving blades. 

,5.3. Tli© Energy Equations 

The equations required for calculating the energy transfer in a process carried out 
in z stages will now be given. In their most general form, these equations may be written 
as follows (the upper sign applies to the turbine, the lower to the compressor) 

--r-c £ H ^ fi. ± + x 0)) 

K ' tr| (!) 

+ h r te)) ?2> 

* The formula and its derivation were contributed verbally by dipl. Ing. A. Eoli. 


78 


2321 









H e T ~-l Ki+h(c,,,) = 

K i=i 



%1 


Hj 

v-W 


(2) 


(3) 


t? . — 

< uk ~ He 


(4) (5) 


2 2 

The term h(u^)( ^ ^ +- 'TT (i)) 111 Equations (l) and (2) is the heat drop which pro¬ 
duces the entrance velocity into the first stage, while the term h(u^))^^fT( 2 )) D 

is that portion of the exit energy from the last stage which may he recovered hy means 
of a proper diffuser. The diffuser efficiency D is so defined that it would he 

unity for a diffuser which recovered the entire kinetic energy of discharge. If D - 
the last term in (l) drops out. 


The question may he raised as to how the exit lose enters into the calculation. 

The kinetic energy which the medium possesses in the last stage is equal to that imparted 
to it at the entrance to the group of stages, increased or decreased by the sum of the 
changes of exit energy in the preceding stages. However, the isentropic changes of 
enthalpy corresponding to these changes are already included in V (from the definition 

of Tj/' ) so that, if no diffuser is provided, only the entrance energy must enter 
explicitly into the equations. The-exit loss, expressed in kcal kg“l , is 


%)( f (2) + 11 (2))(! - D ) 

% 

He is the effective work done at the periphery of the rotor (in heat unite per kg 
of working medium) and u is the corresponding efficiency at the periphery (i.e., the 
"blading" efficiency). The enthalpy differences H^a j, Hy H 0 y in a group of stages 

extending from the first to the j-th stage are given hy Equations (l), (2), and (p). It 
is marely necessary to take the sum up to the j-th stage rather than up to the z-th stage 
and to omit the last term in (l). It is of course necessary to use the value of f which 
corresponds to this group of stages. If the group considered extends from the j-th stage 
to the k-th stage, (l) reduces to 

K 

Haag = —± - hfugi) VJ (6) 

* J ’* 1,) T 


l 


7 ? 


2321 





Equation (2) reduces similarly, "but (3) remains unchanged except that the summation is 
made from the J-th stage to the k-th stage. 

These equations may be written in a more convenient form for practical application. 
It is nearly always possible to assume with sufficient accuracy that the radius r pj j of 
the principal stream surface (at the stage exit) changes from stage to stage by the same 
amount, in which case the values of r pH jform an arithmetic series. If this cannot be 
assveaed in a particular case, at least the stages may be taken in groups and each group 
treated on the basis of this assumption. The modified equations are derived for this 
latter case, which is a very common one, and the simpler case, in which the assumption 
can be made for the entire machine, then follows of itself. The groups of stages will 
be denoted by Roman numerals. 

Consider a group of stages made up of z separate stages, the exit radii of the 
moving blades rgg forming an arithmetic series. Because of the continuity equation to 
be used later, it will be found most convenient to include in this series the entrance 
radius of the first stage (measured to the principal stream surface), which may be con¬ 
sidered as the exit radius of a "zero stage", so that ^(i) z ^2R0 19). If 


A = %(2) " Ml) = r H (2) ‘ r H(l) 


(7) 


%(!) 


H(l) 


then the radius of the i-th stage is 

r 2Hi = r - 


H(l) ( 1 + £ A ) 


Therefore 


£ 

i - 1 2Hi 


r i(D 


i { 1 + A A) ‘ 
H(l) i = 1 V z J 

£ 2 4- (z+l)A -f (z 4-1) (2z 4-1) 


( 8 ) 


Let the expression in the brackets be denoted by Z ( A , z). Then 


2Hi 


rjjd) Z ( A , z) 


(9) 


i = 1 


where Z ( A , z) is to be taken from Table 3. If A = 0 - that is, if the principal 

stream surface is cylindrical - Z = z. An analogous equation holds for ^ u| H1 

Frequently, especially in the case of homogeneous groups or stages, it is suf¬ 
ficient to replace y 1# A i, 7? b i ^ if and 5 R iin Equations (l), (2), and 

(3) by an arithmetical moan value, at least for a group of stages. 


Thus 


Y 




( 10 ) 


80 


232! 




















and analogously for , end . If these simplifications are 

made, the summations In (l), (2), and (3) may he expressed with the help of (9), since 
all the mean values may he taken outside the summation sign. The only summation to he 
made is then 


k. h(u 2 i> = gr Sr 4i 2 ■ -k U (1) Z ( A ’ z > = ' k Wi))z ( A , z) 


i = 1 


i = 1 


For a hlading consisting of groups of stages I, II.IT, with corresponding 

numbers of stages z^., z^ . .the equations then "became 


F ad T 

IC 


it? { h(u(1)l) ^ lZ ( A I'V +h ^(i)n) Y" n z (A n’ z n>+-, • • •} 

± h(u U)l)( f (1) + r (i))+h(u { 2 ) i,)( <p ( ^ f T (!)) D (11) 

/- _ _ 

% r h ( u (i)j) 4* I 7 ? tl 2 P z l) + h ( u (l)n)rii7 tIlZ ^ | 


+ h(u (l)l )(< f (?) + T (i)) 

^ - ^ h(u (l)l) -i~- z ( A I ,z I ) + h(u( 1)n ) 

"Y tl 

+• h (u(i)i)( ^(f) +-T (i)) 


( 12 ) 


^ ii 

^tii 


Z (^ipZ-j-j) 


H eS - ^^(l)!) 1 ! 2 ( ^i>Zj) '*' h (u(i)n)LjjZ ( &u> z ij) 


(13) 


where for a turbine 


L = X(1 - - Y r) 




(14) 


and for a compressor 


L = 


_x_ 

(1 - ? 3p )(l - t e ) 


In spite of the simplifications which have been made, these equations give ouch a 
high degree of accuracy that further refinement would have no significance. The uivision 
into groups is for the most part predetermined by the given construction; that is, the 
groups are not merely ideal abstractions, but actually exist, since it is often not 
possible for mechanical or other reasons to make all stages similar. The accuracy of the 




81 


2321 














calculation may "be increased "by dividing the stages into a greater number of groups, and 
this division may be carried to the point where each group consists only of a single 
stage, so that Equations (l), (2), and (3) would be used for each stage. 

In all these equations it is possible to replace the pressure coefficient 1// by 
the expression 2 /y 2 # where y is the rotor coefficient (see 1.4 (6)). 

In deriving Equations (11), (12) and (13), it was assumed that the individual 
stage groups directly followed one another, so that the exit kinetic energy of one group 
was available as entrance energy for the next, and it was further assumed that this 
entrance energy was Just equal to the exit energy from a ’’zero stage" of the second group. 
A "zero stage" is an additional imaginary stage inserted ahead of the first actual stage 
of a group. Its construction is assumed to be exactly the same as that of the other 
stages, so that it may be considered as an extrapolated stage. It is clear that the 
last stage of the preceding group cannot always be considered as such a "zero stage". In 
such a case additional terms are necessary in the equation for Had to express the 
corresponding additional changes of enthalpy. 

The formulas for the power input or output at the periphery of the rotor, expressed 
in continental hp or kw, respectively, are 


N (hp) = 5.691 GH e » N (KW) * ^* l8 5 GHe 


(14)(16) 


where G is the mass flow in kilograms per second. 

5.4. The Continuity Equation 

If -Q- is the annular area preceding or following any row of blades, and Q is 
the volume flow at this place, then 


.a = 


a 


u 


( 1 ) 


R f 

where y and u are applicable local values. R is a correction factor, accounting 
for the effect of the wall and clearance losses on the flov, which must be experimentally 
determined by a method analogous to that used for the determination of the clearance 
and wall loss coefficients. Erom. energy considerations it has been shown that the total 
wall effect (including the clearance loss) is expressed by the product (l - sv ) 

(l - ~^p), and by exact analogy 



( 2 ) 


Rgp and Rp are given by the following equations. 

Rsp " i f 1 (cr | t*) 


(3) 


1-%= 4J[ 

z 


Y 2 


Y 2 -1 


4- 4 


n 




(4) 


The meaning of f is clear from the form of Equation (3). If the average axial velocity 
through the annular area _Q, is c a , and if the average axial velocity in the clearance 


82 


2321 










“I ® 06 18 °a Bp < 

f ' = <°a ep - °a)A a 

It is evident that if the axial velocity in the clearance space is equal to the mean 
value for the annular area XI , the clearance correction factor R s _ must he equal 
to unity. If c a ^ s c a , f : 0 and : 1. F 

For an acceleration grid, f • >0; for a diffusing grid f» 0. For an ac¬ 
celerating grid with fluid entrance angle ^ q and exit angle oC-^i the order of 

magnitude of f 1 may he determined approximately as follows. It is assumed that the 
flow through the clearance space has the same velocity as the rest of the fluid, hut no 
change of direction; that is, the angle of flow remains oC q, If the contraction 
coefficient of the clearance space is unity. 


f» 


c-j^ sin (A q - c-^ sin <X 1 


c-l sin cA 1 


■ Bin *P . (5) 

sin <X 1 


For diffusing grids the relations are somewhat more complicated, because there is com¬ 
pression in the grid, hut expansion in the clearance space (hack flow). 


Equation (4) is quite analogous to equations 4.10 (3) and (4). The coefficients 
h 8 and h n are the thicknesses of the boundary layers (at the hounding walls of the 

meridian passage, where the velocity is zero) divided by the corresponding blade pitches 
t* and t*, respectively. Equations (3) and (4) could of course he written for the 

stationary blades as well as for the moving blades by adding the distinguishing indices 
' and " to the symbols where necessary. 


In the case of turbines I? 1, 

sp 


and Rp ^ 1, so that 


E = Vr 


iB usually a very good approximation. In the case of compressor for the rows of blades, in 
which +h©re is a pressure increase, Rep < 1, and Rp 1, so that in all cases R 1. 

For practical application it is possible to write Equation (l) in a more convenient 
form. Either the mean diameter Dm or the inside (hub) diameter D^ is ordinarily 
selected as the principal diameter Dg . If the latter is chosen, the inner boundary of 
the meridian passage is the principal stream surface. The choice of has the advantage 

that the relations at the principal diameter are then almost the same as the "mean" 
relations in the stage. 

However, there is also an advantage in using Dn . The design and construction of 
blading which is aerodynamically correct is most difficult at the inside diameter, 
particularly in the case of compressors, where the pressure coefficient of the stage is 
definitely limited by the relations existing at this point. For compressors in particu¬ 
lar, therefore, it is of advantage to select as principal diameter. The annular area 
-CL may then be expressed as 



Also 

u = ^ ^ 

6o 


83 


2321 






where u and D may "be referred either to the Inside or to the mean diameter. If these 
values are now substituted in (l), there is obtained an equation which may be solved for 
any desired variable in terms of the other variables. The most important of these formulas 
will be given. Let 




(6) 

>d ■ / 2 , 

+ 1 

) 2 

(7) 

TC. : Jf+1 

(T 2 -1) 

(8) 


These functions may be taken from Table 2, in which the ratio l/lL is alscj tabulated as 
a function of Y, so that in effect Y is given as a function of l/T^ • Introducing 
these functions, the following equations are obtained. 


(a) Principal Diameter z Mean Diameter 

Um ' 3 - ^Z 


n s 16 925 
* 0.2897 
Da = 0.2897 


Rf (liSf 


f 


Q 

ja. 


Toob V -e 

Yj 8 0.02432 Q 


f 


1000 


A 


'3 


'/3 


f «<P JL Xr \ 


0.02432 Q 


R 


f 1000 


(9) 

(10) 

( 11 ) 
( 12 ) 


(U) 



(b) Principal Diameter 
n . 16 925 

D n = 0.2897 

Xa 


- Inner (Hub) Diameter 



0.02432 Q 

f n 3 
1000 \ 


(14) 

(15) 

(16) 


84 


2321 
















It is to be noted that in these equations the value of ^ depends upon whether E^ or 
is the principal diameter (this is evident from 2.6 (3)). The values of (^ m calcu¬ 
lated for these two principal diameters are connected by the relation 

fm B m - fn E n 

All quantities in these equations are to be expressed in the units given in the Table of 
Notation. The equations may be written for either the entrance or the exit plane of each 
row of blades. 

5.5* Calculation Procedure 

The process of calculating the principal data of a multi-stage turbo-machine by 
use of the given equations will now be outlined. The type of blading is assumed to be 
given, so that the characteristic quantities of the stage are known, and the efficiency 
can be estimated well enough to determine at least the principal data of the machine; 
that is, the speed of rotation, the number of stages, the diameter, and the height of the 
blades. In particular, the quantities \ f J? , Lf? f and of the 

type of stage considered (or of the types of stage if the blading consists of various 
groups with stages of different design) are known. There is no particular limitation on 
the speed. If in practice such a limiting condition actually exists, the first step of 
the calculation - the determination of the speed - is omitted. 

The speed is fixed by either the first or the last stage - usually the last. In 
the case of turbines, the allowable peripheral velocity at the principal diameter of the 
last stage (usually D^), and the permissible radius ratio Y (or the ratio of blade 

height to mean diameter l/E^, which amounts to the same thing) are usually limited by 

stress considerations, the limits depending upon the design of the rotor. In the case 
of axial flow compressors, it is very seldom that mechanical considerations limit the 
design, but if such a case should occur, the critical conditions would be found in the 
first stage. 

Care must be taken that neither Y nor l/E^ becomes too small in the last stage, 
since this would result in very large wall losses. Furthermore, especially in the case 
of blading with a degree of reaction exceeding unity, the tip velocity in the first stage 
should not be so high that the relative velocity approaches the velocity of sound (the 
maximum permissible Mach Number should not exceed 0.9). On the other hand, in the case 
of turbines the minimum permissible value of Y (in this case for the first stage) may be 
decisive. In general, therefore, the stage of lowest pressure determines the highest 
permissible speed, and the stage of highest pressure the minimum required speed (a minimum 
peripheral velocity must of course be specified with due regard to the number of stages). 

The calculation begins with the determination of (from the entrance conditions 

and the final pressure), and of the volume flow Q corresponding to the conditions at the 
different state points (from the thermodynamic properties of the fluid and an expansion 
process with an estimated thermodynamic efficiency t)» 1x1 particular, Q^j and Qr 2 j 

are determined. To obtain the speed of rotation n, 5.4 (9) or (l4) is formulated for the 
stage of lowest or of highest pressure or, in the general case, for both. or 

is then taken from Table 2 for the specified maximum or minimum Y. If sufficient data 
are Lacking, R can be estimated; for the first approximation it is sufficient to take 
r a l. If both possible extreme values of n are calculated in this way, the speed is 
determined within these limits. 

From 5.4 (10), (11), or (15), the values of E^ or E^ at the entrance to the first 
stage and at the exit from the last stage are calculated (it is usually advantageous to 


85 


2321 




calculate Dq "because of ita structural significance). If it is desired to take an even 
value of Djj (ahioh is particularly advantageous in the case of a cylindrical rotor, whore 

Da is the same for all stages), the value selected should differ as little as possible 
from that obtained by the calculation Just outlined. In order to satisfy the continuity 
equation with this selected value, it is then necessary by a reverse calculation to obtain 
for the first and last stages either y_ e or ~)L a from 5 A (13) or (l6), (or X d 

from 5 A. (12)), and to take from Table 2 the radius ratio Y or the ratio l/% - The 
blade heights of the first and last stages are thereby at least approximately determined. 

It will be noted that the entrance annular area is always taken for the first stage, and 
the exit annular area for the last stage. 

From the calculations made up to this point there may be obtained, using the fo rmul a 


u : IT m/6o. 


the tangential velocity at the principal diameter at the entrance to the first stage and 
at the exit from the last stage; that is, u^) and U(2)* E* order to make a first 

estimate of the number of stages, let 


h(u) 


h( u(l)) + h (u( 2 )) 

2 


( 1 ) 


Then 


z 


Had 
V h(u) 


( 2 ) 


If it is expected that there will be different groups of stages, with different values 
of ip , it is necessary to estimate the h(u) of the individual groups from h(u(i) ) 
and h(u( 2 )i;)j an< ^ to write 


Bad = zi M 7 i h ( u i) ■*- z n Y n^( u n) 4 


(3) 


Theoretically there is a possibility of satisfying this equation by different groups of 
values zi, Zjj .... (because the division of the blading into groups is actually 
arbitrary), but for practical reasons the choice to be made is usually quite limited. 
Eventually it is necessary to make a check using the values of u(i)j, u^jj. 

and to repeat the calculation if necessary. From the pressure ratio (P[±]/p[2j) ^ 

(P[2)/P[l]) 'tho machine and the thermodynamic efficiency 'V? the value of f^ 

is taken from Table it-, and the value of f is then found from Equation 5.2 (7), using the 
approximate number of stages Just determined. Equation 5.5 (ll) is then written, the 
efficiency -q °f "th® diffuser which is usually inserted at the end of the blading 

being assumed to be known to a sufficient degree of accuracy. As unknown quantities there 
J* 2 I )> Z( A jj, Zjj 
to satisfy the equation. 


remain Z( A T , z T ), Z( A TT , z-pr ) which again may be selected in various ways 


lor the general case no rule can be given as to the procedure from this point on. 
It is to be borne in mind that in the general case there is a possibility of dividing the 
blading into groups of stages of different designs. The practical case occurring most 
frequently, however, is that in which all stages operate with at least approximately the 
same work coefficient \ , and therefore with nearly equal values of pressure coeffi¬ 
cient • In this case the further procedure is as follows. 


86 


2321 






The division of the stages into groups is first ignored, and the "blading is con¬ 
sidered as if it all "belonged to a single group. T&en only Z enters into 5.3 (ll), and 
since this is only unknown, it can "be-calculated. Since A is given "by 5.3 (7), it 
is possible to find from Table 3 "the number of stages z which, with this value of A * 
gives the value of Z nearest the value Just calculated. Naturally it is very seldom 
that an integral value of z would correspond exactly to the calculated Z. 

The value of Z corresponding to the given A and to the z Just determined 

can now "be substituted in 5.3 (ll), "but one of the other quantities occurring in the 
equation must then "be considered as an unknown, to "be calculated. This might "be \p • 
lowever, a change of (and hence of ^ ) involves a change in all the character¬ 

istic quantities of the stage which are a kinematic nature if the "blade profile and 
position remain the same; in particular, it involves a change of P . Therefore the 
continuity equation must again "be used, preferably in the form 5.4 ( 12 ), ( 13 ) or (l 6 ), to re¬ 
calculate Y for the first and last stages; that is, new blade heights J are determined. 

It is possible,, however, to leave if unchanged in 5.3 (ll) and to calculate a 

new value of h(u^)) • From the new value of u^ follow the new principal diameters 

(that of the last stage being obtained with the help of the given A ), and the conti¬ 
nuity equation 5.4 (12), (13) , or (16) is again used to correct the blade heights. "With 
some practice it is possible to mate a first selection of h(u^)) or of y which does 

not result in exact satisfaction of 5.3 (ll), but which is r ich that the equation will 
probably be exactly satisfied by a second more accurate calc nation. 

On the basis of the data now available, a division into groups is made. Usually 

either a conical or a cylindrical rotor is assumed, so that from the D of the first and 

n 

last stages it is possible to find the at the exit of the last stage of each group, 
and to determine, from 5.4 (13) or (l 6 ), the Y at the exit of each group. All data 
are now available for substitution again in 5.3 (H). Since it is hardily possible to 
change further the number of blades of the individual groups, the equation is now satis¬ 
fied by a change of either the peripheral velocity (that is, of the diameter) or of the 
pressure coefficient. 

' The calculation may now be repeated, with further subdivision of the groups if 
necessary. The better knowledge of the blading dimensions which is now available permits 
substitution of more accurate values of basic efficiency, of clearance loss, and of wall 
loss of the individual stages. If necessary, the thermodynamic basis of the calculation 
may also be improved. Finally, the efficiency at the periphery (the "blading” efficiency) 
is calculated from 5*3 (4) or (5). 

In conclusion, the following remarks may be added. It is known that in the case 
of single stage turbo-machines, such as axial flow compressors, methods are available 
for determining by calculation the optimum proportions (compare C. Keller, Axial Flow 
Compressors). The question arises as to whether such methods might not also be developed 
for use with multi-stage machines. Up to the present time, however, no satisfactory 
basis for such a calculation has been found, and at most it would be possible to develop 
a method• applicable to only one special type of machine. The reason is that the problem 
becomes very much more complex for multi-stage machines. 

In the case of single stage units, the grid loss and the loss in the exit diffuser 
vary, to a certain extent, in opposite directions, and the calculation method used is 
based chiefly on the determination of the proportions necessary to mate the sum of these 
losses a minimum. In multi-stage machines the exit loss is a very much smaller fraction 
of the total loss, so that the grid loss becomes the governing factor in determining the 
most favorable dimensions, and those refinements which were previously neglected now 
become decisive. 

For example, C. Keller in his analysis neglects the effect of blade spacing on the 
drag-lift ratio (glide ratio) of the profile, which is unquestionably penalssible for the 


87 


2321 



very vide blade spacing primarily considered by him. For the much narrower spacing used 
in multi-Btage axial compressors, however, this effect may actually determine the optimum 
proportions. Even apart from this, the determination of optimum proportions would in any 
case be very much more complicated for multi-stage compressors, because the simplifications 
of a purely mathematical nature which Keller has introduced would not be permissible with 
such closely-spaced blades. 

These few remarks will suffice to show why a mathematical procedure for the determi¬ 
nation of optimum proportions has not been given here. 

5.6. Numerical Example: Calculation 
of a Gas Turbine Unit 

Given the problem of determining the speed cf rotation, the diameters, the blade 
heights, and the number of stages of a gas turbine unit (including both turbine and com¬ 
pressor) to satisfy the following specifications. The axial compressor, driven by the 
gas turbine, compresses 20 kg of air per second from an entrance condition p s 1 
atmosphere, t ■ 20° C, to a pressure p - 3.1 atmospheres. The comSpressed air is 
conducted to the point where it is to be used - for example, in a chemical process - and 
returns as gas to the gas turbine inlet, where the conditions are p = 3 atmospheres, 
t *» 500° C. It then expands to a pressure of 1 atmosphere. The gas flow may be 
assumed to be equal to the air flow, and the thermal properties of the gas may be con¬ 
sidered to be practically the same as those of pure air. In a combustion process which 
is carried out with a large excess air ratio, these assumptions are usually valid. 

From these data the values of T and g may be calculated from simple 

thermodynamic relations. If a blading efficiency of 90$ is assumed for the turbine (exit 
loss not included) and 83 $ for the compressor, the temperatures at the exit of the last 
row of blades can be determined, and the inlet and exit volume flows may be calculated 
for both machines from the ideal gas equation. The data are summarized below. 



UctapredSoi' 

Tur'b'ihe 

_2 

Entrance Pressure, p^ kg cm 

1.00 

3.00 

Entrance Temperature, tyj °C 

20 

500 

Entrance Volume Flow, m^ sec " 1 

17.0 

15.1 

-2 

Exit Pressure, p^ kg cm 

3.1 

1.0 

Exit Temperature, tr^j °C 

150.7 

315.3 

Exit Volume Flow, m? sec " 1 

8.01 

3^.1 

Change of Enthalpy, kcal kg " 1 

26.63 

50.3 

Mass Flow, G kg sec"-*- 

20 

20 


It will be assumed that the blading for both compressor and turbine is such that the 
characteristic numbers are the same for all stages at the principal diameter, which 
is taken as for the compressor., and Ifo for the turbine. The characteristic numbers 
are tabulated below (see also the velocity triangles, Figure 20). 


88 


2321 









Compressor 

Turbine 

p 

0.50 

0.50 


0.35 

0.40 

X, 

0.395 

- - 

■tz 

0.605 

-0.037 

A 

0.420 

2.148 

lo 

0.92 

0.95 

Y 

0.386 

2.26 

R 

0.95 

1.00 

0 u/c]_ 

• 0.9, 

OJ 

11 


The data far the turbine correspond to u/cj_ 

and R are estimated. The stages are otherwise assumed to have "correctly" shaped 
"blading; that is, they are considered to he normal stages (cf. 2 . 7 ). 


(a) Determination of Speed of Rotation . 


The minimum speed of rotation required is determined hy the stage with the smallest 
volume flow; that is, the last stage of the compressor. In order that the number of 
stages may not "became excessive, Un will he taken as at least l 60 meters per second, and 
in order to ensure reasonable efficiency, the radius ratio Y will he taken as at least 
1.2. Equation 5.4 (l4) written for the exit of the last stage with these minimum values 
(i.e., with Xa * 0.44) becomes 

n^ = 16 , 925 | 0*95 * 0.35 ^ 4.1 * 0.44 j 2 - 4, 620 min” 1 

The greatest possible speed which can he selected is determined hy the stage with the 
greatest volume flow; that is, the last stage of the turbine. For mechanical reasons 
cannot he permitted to exceed 240 meters per second, while Y should not exceed 1.6. 

With these maximum -values 5.4 (9) becomes 


= 16, 925 1 • 0.4 • 13.85 • 0.922 


“max 

The speed will be selected as 

(h) Compressor Dimensions. 


34.1 

n - 6000 rpm 


Let the radius ratio Y in the first stage he about 1.5. Then 
that from 5.4 (15) 


Dn = 0.2897 


J&SL 


0.95 * 0.35 • 6 


• 1.25 


6 , 550 min 


-1 


la = 1.25, so 

0.549 m 


Let Lj! a 0.550 meters, without recalculation of Y, since the value R = 0.95 is 
merely estimated. Let the rotor he cylindrical, so that - 0.550 for all stages. 

Equation 5.4 ( 16 ) shows that ^)C a is simply proportional to Q, so that for the last 
stage 


89 


2321 













> 


X A r 1.25 ■ fcp j k- ■ = °*589 

a 17.0 

From Tall© 2, Y - 1.26. For Ih . 0.550 and n 2 6000, 

% = - 6000 . 172.7 m sec' 

60 

k(un) - 172.7 2 - 5.56 local kg" 1 

91.5 


Substitution can now bo made in 5»3 (ll). T 1 is also to be substituted for (2)> 

since with the specified blading the stationary grid "follows" the moving grid (in the 
sense that the last row of blades is a stationary row) Let p = With ^^ = 

q Z' 0.86 (estimated), and Pf2]^ p [l] " 3* 1 # it follows from Table 1+ that 

f s 0.027. Since the number of stages required can be estimated as approximately 20, 

0.026 

3.56 (0.l2r<? t 0.156) + 3.56 * 

• (0.1225 +• 0.156) 0.3 


Equation 5.3 (ll) then becomes 

26.63 = (3.56 • 0.386 






From this equation z > 20.4, so that the number of stages can be taken as z - 20. 


The compressor calculations are carried no further for the reasons given under 
(d). The principal compressor data obtained from this preliminary calculation are tabu¬ 
lated below. 



Entrance 

Exit 

Inner (Hub) Diameter, 

mm 

550 

550 

Outer (Tip) Diameter, D s 

/ 

mm 

825 

693 

Blade Height, 1 

mm 

^7.5 

71.5 

Rotor Velocity at Inner 




Diameter, 

m sec -1 

172.7 

172.7 

Rotor Velocity at 




Outer Diameter, u s 

m sec -1 

259 

217.4 

Number of Stages, z 


20 


* 




90 

2321 











(c) Turbine Dimensions 


At the exit the radius ratio Y will be assumed to be approximately 1.6. Ercm. 
Table 2, Xe = 2.03, and from 5.4 (ll) 


\ - 0.2897 


34.1 


1 


0.4 


2.03 


1/3 


0.554 m 


A cylindrical rotor will again be assumed with - 0.55 % so that Y must be cor¬ 
respondingly corrected. From 5.4 (13). 


X 


e - 


0. 02432 • 34.1 

0.4 • 5 


2.073 


. Y = i.6l 


Z • 0.1667 

This value of Y for the last stage is still permissible (although origina l ly 1.6 had been 
assumed as the highest permissible value), because the peripheral velocity is somewhat 
lower. As In the case of the compressor, it is evident from 5.4 (13) that for a cy¬ 
lindrical rotor X Q is proportional to Q, so that the radius ratio Y at the entrance 

is determined. 

X e = 2.073 15a = 0.918 .•. Y * 1.337 ^ 1.34 

3 ^ • 1 . 

The mean diameter can now be calculated at both entrance and exit# 


% - Y -f 1 

2 

D n 

The corresponding rotor velocity follows directly. 

Entrance 

Exit 

- 0.6425 m 

D m - 0.7175 m 

3 202.0 m sec" 1 

% » 225.5 m 1 

h(V = 4.87 kcal kg" 1 

h(u Jn ) - 6.08 kcal 

From 5.3 (7) 


A : 0.7175 - 0.6425 

s 0.1167 

0.6425 


The number of stages is estimated from 5.5 (l) and (2) 


r 1 

-1 


E(%) 


4.87 6.08 

2 


5.48 kcal kg -1 


z ^ 50.3 

2.26 • 5.48 


4.06 


Assuming z = 4, it is found from Table 3 that Z . 4.6l. The heat recovery factor (l +- f) 
_ 1.0075. Substitution is now made in 5*3 (ll), which is to be solved for -\p . 


91 


2321 













50.3 = 1 14.87 V * 4.6l ( +- 4.87 (0.16 + 0.0014) 

1.0075 L J 

\j/ = 2.22 

Originally it was assured that = 2.26. The agreement is therefore very good so 

far, and if the pressure coefficient were about 2$ lower, the calculated results would 
he consistent with the assumptions. It must he noted, however, that the error which was 
introduced hy disregarding any division Into groups has changed the relations somewhat. 

The diameter at the exit of the second stage is actually somewhat smaller than the 

arithmetical average of Dm(i) and Dd(2) (since the gas volume flow Increases progressively 
in the course of the expansion), so that a more accurate calculation would result In a 
value of vp somewhat greater than 2.22. 

The-results already obtained, which are nearly exact, are summarized in the 
following table. 



Entrance 

Exit 

Inner (Hub) Diameter, D n 

mm 

550 

550 

Mean Diameter, 1^ 

mm 

642.5 

717.5 

Outer (Tip) Diameter, D B 

mm 

- 735 

885 

Blade Height, 1 

mm 

92.5 

167.5 

Rotor Velocity at Mean 




Diameter, u^ 

m sec - '*' 

202.0 

225.5 

Number of Stages, z 


4 



(d) Remarks . 

In the case of both compressor and turbine the calculation has not been carried 
further than the first approximation (in itself very good) because this is sufficient 
to illustrate the method. As a rule, however, even in practical cases the calculation 
would not be carried further until the details of construction have been worked out to a 
point where there is a more reliable basis for refined calculation (taking into con¬ 
sideration leakage through the shaft packing and the pressure equalizing devices, more 
exact specification of the clearances, etc.). A more accurate estimate of the stage 
efficiency can then be made. 

5.7. Us© of Dimensionless Variables in 
Multi-Stage Blading Analysis 

The phenomena occurring in one of the stages of a multi-stage system of blading 
can be satisfactorily determined only by tests of multi-stage machines. In single stage 
tests it is not easily possible to simulate the entrance and exit flow conditions which 
actually exist in a multi-stage machine. If tests are made on a group of middle stages 
of a multi-stage system with identical individual stages (eventually perhaps on a single 
stage only), these stages within rather wide limits may be considered as very nearly 
homogeneous. This is true even if the test is made with a pressure ratio p^j /v [2] other 

than the design value, since the disturbance introduced is noticeable only at the exit 


92 











end of the "blading, and particularly in the last stage 


In Section 4.4 it was found that the ensemble of possibilities of matching two 
given "blade grids in a homogeneous state element is singly-parametric. This applies also 
to homogeneous stages, so that the operating condition may "be characterized "by specifying 
a single characteristic quantity (however, this is a first approximation only, as ex¬ 
plained later). The characteristic quantity should "be so chosen as to "be conveniently 
measurable; the pressure coefficient yj , for example, would satisfy this requirement. 

If is selected, it is merely necessary to measure p^, and p^ of the group 

of stages to "be investigated; can then he calculated, and Equation 5.3 (6) can he 

written. From this equation ~y/ may "be calculated (the number of stages in the group 
must he small enough so that "\^ can he assumed to he the same for each), since uq ± 

is known from the measured speed and the design data. In theory it is best to test only 
a single stage in the middle of the multi-stage blading, hut the procedure outlined will 
give a very high degree of accuracy. 

The rotor coefficient V (definition. 1.40 0), which is equivalent to the pressure 
coefficient , might of course he selected as the quantity characterizing the 

operating condition. In the case of turbines this may he particularly desirable because 


by a parabolic 


the efficiencies ^ o > 



type of curve which, for simplicity in a elementary treatment, may replace the actual 
efficiency parabola. 

Strictly speaking, the operating condition of a given blading is not characterized 
completely by either i|) or |/ , because the ensemble of the possibilities of matching 

two given grids in a homogeneous stage element is singly-parametrio only when the effects 
of Reynolds Number and of Mach Number are ignored. In cases where these effects cannot 
be considered negligible, the ensemble has three parameters, and the test results are 
basically functions of three dimensionless quantities; i.e., or )? ^ Reynolds 

Number, and Mach Number. 


93 


2321 



VI. THEORY OF THREE -DIMEIEIQNAL FLOW OF 
RON-VISCOUS FLUIDS THROUGH TUKBO- 
MACHIRE STAGES 


6.1. Notation and Preliminary Remarks 

The notation given In Sections 1.3 and 2.1 is also used in this chapter. The 
symbols A, G, K, and y have somewhat different meanings, but the differences are such 
that no confusion can result. The signs of the characteristic quantities of a stage are 
defined in the same way as in Chapter U; for example, \ is negative for compressor 
stages. The following additional notation is used. 


x, y, & Dimensionless cylindrical coordinates (see Figure 22) 
1, k Unit vectors in the coordinate system x, y, & 


A, A lt Ag 
Bj Bp 


Coefficients in various formulas 


C b Ca 
1 = la 
E, A E 


F 


F, G 

J 0> J 1 
% 


Or J. 4- Cu £ Characteristic velocity (See 6.4 (b) (l)) 

i 4-Q-r 1 Characteristic superposed velocity (See 6.4 (7)) 

Mechanical energy content, change of mechanical energy content 
[m 2 sec"^], but dimensionless in 6.6 

Field Force per unit mass £m sec'^J 
Functions in the solution formula 6.4 (21) 

Bessel’s cylindrical functions of orders 0 and 1 
Neumann’s cylindrical functions of orders 0 and 1 


Zo# ^l 

Q* 


General linear combination of Jq and N 0 , of and N^ 

Dimensionless expression (obtained by combination with charac¬ 
teristic velocity) for the volume flow. 

Tangential component of characteristic relative velocity 
Integral defined by 6.6 (28) 


a Axial length of stage divided by inner (hub) radius 

kj_ i-th characteristic value of the problem treated in 6.4 


Ho*, P i* Dimensionless expression (obtained by combination with charac- 
’ teristic velocity) for circulation preceding, following stationary 

blade grid 


A t 


Normal value corresponding to the i-th characteristic value, as 
defined by 6.4 (37) 

Potential of the superposed velocity 


94 


2321 





Stokes* stream function [ m? sec“^J , "but dimensionless in al3 
relations following 6.4 (42) 

Constant in the expression for Zq (see 6.4 (27) and (30)) 

Angle of inclination of stationary blade (considered as a vortex 
filament) with respect to the radius. 

In this chapter the common laws of hydrodynamics will "be assumed (as given, for 
example, in Mine-Thomson*s "Theoretical Hydrodynamics"). Details of the cylindrical 
functions used in 6.4 and 6.5 will "be found in Me Lachlan 1 s "Bessel Functions for Engi¬ 
neers". "Cross flow" in this chapter will "be understood to mean flow In a radial 
direction. 

6.2. General Survey of the Phenomena 

In the previous chapters the calculation of axial flow turbo-machines has be6. 

"based on assumptions which very much simplified the actual meridian streamline pattern. 

In general, it was assumed that the flow is along cylindrical surfaces, and that the 
axial component of the velocity does not change along the radius. The actual flow may 
"be so different, however, that it is desirable to improve the theoretical "basis of the 
treatment if possible, and the analysis here given is intended as a contribution to 
this end. 

Consider the flow of an incompressible fluid through an axial turbo-machine stage, 
the meridian passage of which is bounded by cylindrical walls (so that it is a "purely 
axial" stage). It follows that: 

(a) If the blade pitch is infinitely snail', and if the shape of the blades is 
suoh that both the axial velocity component and the velocity moment (tangential component 
multiplied by radius) are constant along r, there can be no cross flow for a friction¬ 
less fluid. The stream surfaces are in fact co-axial Cylinders, and the assumption 

of constant axial velocity involves no contradiction, since, such a flow is in a con¬ 
dition of equilibrium. A stage of this kind, is a "normal stage" (cf 2.7) in the highest 
possible degree. 

(b) If conditions are otherwise the same as for (a), but the blade pitch is 
finite (so that the assumptions can refer only to the mean value of the velocity along 
the circumference), there will be a small cross-flow. In the general case the mean 
value of the radial velocity along the circumference does not vanish, so that on the 
average there will remain a "resultant cross flow" (cf 4.2 also). 

(c) If conditions are otherwise the same as for (b), but the fluid is not friction¬ 
less, there will be an additional cross-flow, since the boundary layers on the moving 
blades are subjected to a centrifugal field of force, and secondary flows occur by reason 
of the curvature of the stream. 

(d) If, with infinitely small pitch, the variation of blade shape along the 
radius is arbitrary (for example, if the blades are not "twisted"), there will be cross- 
flow even with frictionless fluids, and the axial component of velocity will not be 
constant along the radius. The assumption of flow along cylindrical surfaces with con¬ 
stant axial velocity component then contradicts,-in general, the basic equations of hydro¬ 
dynamics. 

(e) If the conditions are otherwise the same as for (d), but the blade pitch is 
finite, there will be additional cross-flow as under (b). 

(f) If the conditions are otherwise the same as for (e), but the fluid is not 
frictionless, the same type of cross-flow mentioned under (c) will occur. 


A 


95 


2321 




1 


(g) Regardless of any other assumptions, cross-flow is always to he expected if 
the stationary blades are not exactly radial, hut are sloped even slightly with refer¬ 
ence to the radial direction. 

(h) If the restriction as to incompressible fluids is removed, in general at least 
one of the boundary walls of the meridian passage will not be cylindrical, and this of it¬ 
self is sufficient to cause cross-flow. However, even if both boundary walls are cy¬ 
lindrical, "purely axial" flow (that is, flow along cylindrical surfaces) can result in 
no possible condition of equilibrium if the fluid is compressible. 

In this chapter the effects mentioned under (d), (g), and (h) will be discussed. 

It is Inown that the variation of blade shape along the radius for "correctly" shaped 
blading - that is, for normal stages - is very great. The equations of 2.7* which give 
the variation of the most important characteristic quantities along the radius, show 
that there is a very great difference of flow angle (and consequently of blade shape) 
at different radii, especially for rotor blades. If in practical cases a shape which is 
the same at all radii is substituted for the theoretical shape (as is usually the case in 
steam turbine practice), the departure from the "theoretically correct" case of the normal 
stage is so great that large departures are to be expected from the condition of no cross- 
flow which obtains with constant axial velocity along the radius. One instinctively feels 
that such a great simplification must necessarily resulo in a correspondingly large dis¬ 
crepancy, and this subject is therefore investigated more thoroughly in Section 6.6. 

The proposal that the stationary blades be slightly inclined, rather than exactly 
radial, has been made several times. The purpose has usually been to reduce noise (in 
the case of compressors), but the proposal has also been made on purely aerodynamic grounds 
(cf. Darrius, Stodola Anniversary Volume). In Section 6.h an investigation ie made to 
determine how this method of constructing stationary blades affects the flow, but only for 
the case when the blading is designed in accordance with the requirements of the potential, 
theory. 

Finally, the effect of compressibility is discussed in Section 6.8. The exact 
treatment is extremely complicated, particularly because this effect cannot be isolated, 
but must be considered in connection with other effects which are already very complicated 
in themselves, and the analysis given in Section 6.8 is therefore limited to a few 
fundamentals. However, the question of compressibility is of no great moment for those 
cases to which the present theory is particularly applicable, in which the net charge 
of density per row of blades is very small. 

6.3 6.3. General Discussion of the Problem 

In this section consideration is limited, to frictionless incompressible fluids, 
since the phenomena of paragraphs 6.2 (d) and (g) are the subject matter of investigation. 
The hydrodynamic equations which govern the motion of such a fluid are: 


+ f£v)c =£- rj-vp 

7)t u 


( 1 ) 


V £ = 0 


( 2 ) 




Equation (l) expresses in vectorial form Euler’s three equations of motion, and (2) is 
the continuity equation. Since a flow process non-variant in time is considered, the 
partial derivative with respect to t vanishes. By a purely formal transformation. 
Equation (l) may be written 


c x rot c 


- £ -h grad 


P 



(3) 


96 


2321 




where P 8 p^i . F is a field force ■which may under certain circumstances "be sub¬ 
stituted for the "blade forces in the theory of turbo-machines. 

Consider a single stationary row of "blades. The flow Is bounded^ on the inside and 
outside "by cylindrical walls which, as a first approximation, may "be considered to extend 
to infinity in "both directions (see Figure 21). Let the stationary "blade "be so designed 
as to impart to the flow an exit angle dX. # which is a function of the radius only 

and may "be arbitrarily specified. Under these conditions, if there is potential flow on 
the upstream side of the "blades, there cannot in general "be potential flow on the down¬ 
stream side. This may "be demonstrated as follows. 

Assume that the row of stationary "blades is replaced "by a vortex filament c as 
shown in Figure 21. If there were a potential flow on the downstream side of the "blades, 
then everywhere in the region considered (to the right of the "blade in Figure 21) the 
law o^r s constant must hold, regardless of what the meridian streamline pattern may 

"be, since otherwise there would "be an axial vortex component present. But if c^r - con¬ 
stant, the variation of the axial component at the exit is also fixed, "because c fi ■ c^ 

tg , , and (X ] ls specified as a function of r. Therefore o a and c u in the plane of 
the grid are completely determined (except for a constant factor which is not important 
for this problem). 

Moreover, since the normal component of velocity is zero everywhere at the inner 
and outer wall boundaries, it may be considered that there is a region of space extending 
to infinity in which at the finite boundary surfaces a, ’b, and c (Figure 21) the normal 
component of the velocity is given. Therefore the non-cyclic portion of the potential 
flow is determined, and in view of the relation o a /o n ^ tg o( j which exists in the 

plane of the grid, the cyclic portion is also determined. It is therefore clear that, . 
for a given variation of (X , with r. the potential flow on the downstream Bide of, the 
blades (if such a flow actually exists) is determined completely except for a constant 
factor. 


The same analysis applies exactly to the upstream region, for the reason that here 
also the normal component is determined everywhere at the bounding surfaces, since by 
continuity the a/H al component c a at the grid plane must be the same as for the down¬ 
stream region. It is still possible to specify arbitrarily the cyclic portion of the 

entrance flow, but this is Immaterial for present purposes. 

It may be concluded from the’ preceding analysis that the meridian streamline 
pattern is not only uniquely determined for both regions, but is the same for both. 

That is, the pattern is symmetrical, so that the stationary grid plane is a plane of 
symmetry. It follows that in general the stream lines in this plane are bent in such a 
way that the radial component' c*. changes sign. This discontinuity is absent only when 
the tangent to the stream line at x = 0 (grid plane) is parallel to the x axis (axis 

of rotation), so that Cj, = 0. Mere exact calculation shows that this can be the case 

everywhere along r only if c a is everywhere constant, so that the stream lines are all 
straight lines (cf. 6.4). This requires, however, a very special variation of d 

along r; namely, such a variation as satisfies the equation. 

r ctg o( 2. = constant 

and this "is exactly the case of "correctly” shaped blading. 

For every otner given variation of angle there must be a radial velocity in the 
grid plane if the flow is to be essentially of the potential type on the downstream side. 
As a practical matter this means simply that the stationary blades must deflect the fluid 
in a radial as well as in a circumferential direction, which may be accomplished through 
t he use of additional guiding means, or more simply (in accordance with Darrieus* 


97 


2321 


suggestion) "by designing the stationary "blades with a slight angle of inclination to the 
radial direction. Strictly speaking, this angle mast "be a definite function of the 
radius, so that the "blades are "bent in a specific fashion, somewhat as illustrated in 
Figure 25. If, with specified angles,the "blades are not so shaped that they can deflect 
the fluid in a radial direction, there cannot "be potential flow on the downstream side 
(as has already "been pointed out), since the condition of a continuous radial velocity 
in the grid plane must still "be satisfied, and this condition cannot "be satisfied with 
potential flow except when r ctg \ a constant. The results of this analysis may 
therefore "be summarized as follows. 

If the flow on the upstream side of a single stationary "blade grid (the assumed 

arrangement' "being as shown in' Figure '21)' is of the potential type, and if the "blade is 

so smvped’ 'tlmt 't£e exit angle <x. i varies”"vit!h the radius in accordance with a specified 

law,' 'tho'ro 'cannot' 'ini general' b'e potential x'low on. the "downstream side." ' If 'such.' a ' ' '' 

potential flow 'is desired,' it is necessary to design the "blades to deflect the fluid in 

a radial as well as In a circumferentlal direction, the amount of the deflection "being 

dependent upon the prescribed variation of cd \ 9-long r. Only when r ctn q(^ z 

constant' is no~radlal' deflection required, the 'streamlines in 'th'ls' case lying on 

^ylindrloai 'surfaces and ijie axial velocity 'being constant. ' 1 .. . ‘ 

The preceding analysis was "based on the" special configuration shown in Figure 21, 
"but it is valid also for the general case. In particular, it is applicable even when the 
boundary walls are not cylindrical, and when other boundary conditions exist by reason 
of the presence of a grid of moving blades a short distance on the downstream side of 
the stationary blades. In such cases the relations are much more complicated, but usually 
a very special design of stationary blades is necessary if there is to be potential flow 
on the discharge side. 

The very simple arrangement which served as the basis for the preceding analysis 
can often serve as a first approximation to the actual case. This is particularly true 
if the moving blade grid following the stationary blades does not affect the meridian 
streamline pattern - a condition which exists when the moving blades are so designed 
that, for the given meridian streamline pattern, the angle of the absolute velocity of 
the fluid varies in such a way that rc u is not a function of the radius. Basically, 
the analysis here carried through for the stationary blades applies also to the moving 
blades (using absolute velocity), except that for reasons of mechanical strength the 
latter can probably not be designed to exert a radial deflecting force. 

The creation of vortices in a flow through stationary passages which was originally 
vortex-free does not contradict the Helmholtz vortex theorem, and is explained by the 
well-known Kutta trailing edge condition of the airfoil theory. For viscous fluids it 
is known that the rear stagnation point lies at the rear edge of the blade profile, and 
it remains at this point if the viscosity is allowed to approach zero. This condition 
determines the circulation around the profile, and since in general the circulation 
varies along the blade, corresponding vortices are induced. This is the basic reason 
why there cannot be potential flow on the downstream side of the blades. If the 
stationary grid is replaced by a field of force, the origin of the induced vortex is 
explained by the fact that in general this is not a potential field. 

If flow non-variant in time is not a potential flow, the total hydrodynamic 
pressure energy (P c,/2) is constant along every stream line, but in general the value 
of the constant will be different for different streamlines. If the flow through a 
stationary blade grid with potential flow on the upstream side is considered, a special 
characteristic will appear. The course of some single streamline will be traced,* 
beginning in the region on the upstream side, passing through the blade and ending in 
the region of the downstream side. Along the entire path the total pressure energy 
remains everywhere constant, including the path through the blade itself, which is 
merely a curved duct to guide the fluid without delivery or extraction of energy. 

Even if a stationary blade grid is considered to be replaced by a field of force, this 
does not change the total pressure energy of the fluid, since the field is so constituted 
that it does no work on the fluid, and the fluid does no work against the field. 


98 


2321 
















Since for potential fl,ow the hydrodynamic total pressure energy is constant not 
only along every stream line, hut over the entire field, and since it is not changed hy 
the passage of a streamline through a stationary blade grid, it follows that this pressure 
energy is also constant everywhere in the region on the downstream side of the blade. 

The term grad (P +- c 2 /2) appearing in Equation (3) is therefore zero in this region, 

and since there is no field force, 

£ i rot £ 8 0 (!<■) 

This equation is satisfied if rot £ - 0 (that is, if the flow on the downstream side 
is also a potential flow) or if the vector rot £ at every point has the same direction 
as £ itself. The following conclusion may therefore be drawn. 

If there is potential flow cn the upstream side of a stationary blade grid, but not on 

the downstream side, the velocity rotation veotor at every point has the same direction 

as the velocity vector, and the hydrodynamic total pressure energy is consequently every¬ 

where the same. 


This conclusion cannot be applied to a moving blade grid, since in this case 
(P •+• c 2 /2) changes because of the gain or loss of energy by the fluid. The magnitude 
of this energy transfer is 

Ae - CO(r lGul - r 2 c u 2 ) ( 5 ) 

P z 2/n (r^c^ - r 2 c u2 ) iB ‘ fche Bum °? circulations around all moving blade 

profiles in the stream surface considered. A E is equal to the change which the total 
pressure energy (P •+• c 2 /2) undergoes in the moving blade grid. At the exit of this grid 


dr. 


2 + 


d 

br 2 


A E : - 


u> 

2 7T 


( 6 ) 

or 2 


The minus sign indicates that (P2 +• c ? / 2 ) is decreased by the amount A E. For 
compressors ^ E is Itself negative. Equation ( 6 ) gives the radial component of grad 

(P 2 -f- Cg 2 / 2 ), which is therefore not equal to zero so long as P is not constant along 


the radius. For infinitely small pitch, this gradient has no tangential component, be¬ 
cause the total pressure energy along the circumference of the same stream surface does 
not change. Therefore, except for the region inside the rotor blading itself (or except 
for the field of force which replaces the moving blade grid), the streamline surfaces 
are surfaces of constant total pressure energy, and the vector grad (P 2 f. c 2 2 /2) is 


at every point perpendicular to the corresponding stream surface. This veotor has in 
general an axial component, but this is very much smaller than the radial component and 
can often be neglected. Equation (3) written for the region on the downstream side of 
the moving blades then becomes 


£2 x rot £2 = grad (? 2 + eg 2 ] ( 7 ) 


Since £2 is tangent to the stream surface, and the gradient on the right-hand side of 
the equation is perpendicular to the same surface, rot £2 must lie in the same tangent 
plane as £2 itself, as illustrated in Figure 22. To summarize. 


99 


2321 















If the total pressure energy (P -f~ c 2 /2) in the region on the upstream side of 
a moving blade grid" Is everywhere the same, in the region on the downstream side it will 

in general he different from stream surface to stream surface, the variation “being a _ 

function of the change of p in accordance with Equation (o). p' g and rot cp lie in the 

tangent plane to the stream surface at the point considered, and grad (P 2 +- c 2 2/2) is 

perpendicular to this plane. 

Considering now multi-stage blading, at the entrance to the second stationary row 
the value of (P 2 -f- c 2 /2) is different for the different stream surfaces, as was shown 

by the previous analysis, but it remains constant for the same surface up to the entrance 
to the moving row. In this row the total pressure energy is again changed by the amount 

A E r P y which is different for the individual stream surfaces so long 

as P is not the same. The configuration at the exit of the second moving blade 
grid is qualitatively exactly the same as at the exit of the first; only quantitatively 
is there, in general, any difference. Thus the same analysis may be repeated for the 
third stage, and so on. 

Since for every stream surface the changes of total pressure energy are additive 
from stage to stage, the differences between the total pressure energies of the individual 
stream surfaces are also additive. It is advantageous at this point to introduce the 
Stokes* stream function ■Or , defined by the equation. 


aJL > °r = - 1 d (8) 

Or r 5“x 



It may easily be shown that the function defined quite generally in this way for ro¬ 
tational symmetry of flow is constant for every stream surface, and that the volume 
flow between two stream surfaces (denoted by the subscripts a and b) is 

Vb * 211 ('f'b * f a > (9) 

If a flow with rotational symmetry is specified, therefore, the function V is fixed, 
except for an additive constant which is not important for the present analysis. 

Since every stream surface may be characterized by a value of ■qr associated ’ 

with it, there must also be a relation between and P for a given row of 

moving blades. Furthermore, there must exist in every portion of the region between two 
successive rows of moving blades a unique functional relationship between ^ and 

(P 1“ c 2 /2), Just as Equation (6) was obtained by differentiation of (5) with respect 
to T 2 , the following equation for the exit of the first stage may be obtained by 
differentiation with respect to "iTf 


a 

£>t 



d 4 E - - J*L 

dip " 2ir ajr 


do) 


It has already been stated that for the same stream surface, and therefore for the same 
o£r > "tt 1 ® values of A E are additive from stage to stage, so that after the 

k-th row of moving blades the total change of energy has the value 


k 
£ 
i = l 


&£{( f) 



(ii) 


100 


2321 















The gradient of the total pressure energy on the downstream, side of the k-th row of 
moving blades is therefore 




—, £ Ati = 

i=i l 


- — % 
2ir t 



( 12 ) 


Consider the case of frictionless flow of an incompressible fluid through a system 
of blading consisting of a large number of exactly similar stages (Figure 23). The con¬ 
ditions in the second stage will not be exactly the same as in the first, the conditions 
in the third stage will depart somewhat from those in the second, and so on, but the 
differences will soon become very small. In successive stages there is approached 
asymptotically, but very rapidly, a condition of flow which simply repeats itself periodi¬ 
cally. The last stages show an analogous behavior. That this actually occurs can be 
demonstrated in the following manner. 

The stream conditions on the discharge side of the first row of blades are determined 
by the specified volume flow Q and by the conditions on the upstream side. A similar 
relation holds for the second row, and it is apparent that the conditions on the discharge 
side of the second row depend relatively little on those preceding the first row, since 
the effect of the latter will be hardly noticeable here. The conditions preceding the 
second row of blades depend only in part on those preceding’ the first, and therefore those 
on-the discharge side of the second row depend in still smaller measure on those preceding 
the first row. 

It follows as a logical consequence that the conditions on the downstream side of 
the second row of moving blades are very similar to those on the downstream side of the 
first row. The conditions on the downstream side of the third row are very similar to 
those on the downstream side of the second, since the initial conditions for the third 
stage differ very little from those of the second, and the effects of even these small 
differences are greatly reduced on the discharge side. There is thus approached asymp¬ 
totically a flow condition which is periodically repeated. 

A mathematical measure of the extent to which a given initial condition is "carried 
through" a give blading for a given Q and 60 is the degree of dependence of the dis¬ 
charge angle on the entrance angle for the particular grid used. This dependence is 
relatively small in the case of those grids chiefly used for multistage machines, and 
particularly for accelerating grids. It follows that for turbines, in particular, the , 
periodic condition is reached very rapidly. For all practical purposes it may be assumed 
that it is already attained in the second stage. 

Without further proof it may be concluded from (12) that, lor such an asymptotic 
flow condition, the relation 

if-o 

d f 

must hold. Were this not so, d (P 0 c 0 2 /2) must be assumed to increase in 

d IJr 

magnitude from stage to stage. This, however, would contradict the periodicity of the 
process and lead very quickly to impossible energy relations. The meaning of df/cJilf-0 

is simply that the circulation is constant along a moving blade. From the periodicity 
of the process (and therefore from the fact that the stages may be considered to be 
homogeneous), it follows that the circulation along a stationary blade is also constant. 

It must be strongly emphasized that this is valid only if, in the calculation of 
circulation, the paths of integration lie on stream surfaces, and are not arbitrary 
curves in spare which may intersect those surfaces. 


101 


3321 






The results of this analysis may he summarized as follows: 


If a frictlanless incompressible fluid flows through a series of successive _ 

identical stages, in each Individual stage the conditions of flow will he the same to a 

very high degree of approximation, so that the flow relations simply - repeat themselves 

periodically. Only in the first few and in the last few stages (chiefly only in the 


first and last) will Idle conditions be 'appreciably different. The periodic process 

in the ^middle" stages depends on the volume flow and on the speed of rotation; it is 

characterized by the fact that the circulation Itatoan on stream surfaces) around the 

blade is constant for loth stationary and moving rows. The flow is not a potential 

flow unless every Individual row of blades fulfills -die required conditions. 


In concluding this section, the idealisation involved in the assumption of infinitely 
small blade pitch will be further examined. Consider, for example, a blade grid in which 
the circulation Is not constant along the blades, so that vortices are induced with un¬ 
stable surfaces leaving the exit edges. The flow on the downstream side consists, in an 
exact sense, of "layers" within which there is a vortex-free flow, and which are separated 
from one another by these unstable surfaces. The structure of this flow therefore differs 
completely from that of a steady flow with vortices, and remains different no matter how 
small the pitch of the blades. The question then arises as to whether, in the limiting 
case of very small pitch, the transition to continuous vortex distribution is permissible. 
This limiting transition can be discussed by means of an exanple which is much simpler 
than the case presented by flow discharged from a grid of blades. 

Figure 2b shows cases of simple parallel flow between two plane walls. In case (a) 
the flow consists of two "layers" separated by an unstable surface, and in cases (b) and 
(o) of four and six "layers", respectively. Case (d) shows continuous vortex distribution 
with no surfaces of discontinuity. Although the structure of the flow in (d) is completely 
different from that of (c) an engineer will without hesitation substitute one for the 
other by way of approximation, and he would consider the flow in (d) as the limiting case 
approached by (a), (b), and (c). That this is fundamentally sound can be verified in the 
following manner. 

The motion of a frictionless fluid is determined by the equations of motion (momen¬ 
tum law), the continuity equation, and the boundary conditions. If the flows of cases 
(a) and (b) are examined at a certain point A which exactly corresponds for the two cases 
(Figure 2b ), the velocities will differ only very slightly. Consequently the momenta of 
two particles of fluid of equal mass which are momentarily at this point will also differ 
very little. By sufficient subdivision into "layers", the differences of velocity and 
momentum can be made as small, as desired, and at corresponding points of flows (c) and 
(d) the velocities and momenta can therefore be made to differ from each other by as 
small an amount as desired. This is true not only for the specified type of flow, but 
also for flow in general. 


From the equations of motion and of continuity It follows that the dynamical con¬ 
ditions can in the’ same way be made to differ by only a very snail amount. If for 
example, a body (such as a sphere, or the lihe) were to be inserted between the two * 
boundary planes of (c), and another similar body were similarly inserted In (d), the forces 
acting on these bodies would be very nearly the same, and the effect of the body on the 
flow would be very nearly the same, in the two cases. The flow conditions by sufficiently 
fine subdivision could be made as nearly identical as desired; therefore-the limiting 
transition is Justified. 

The fineness of the "layers" into which the flow in a normal turbine blading is 
divided may easily be visualized. Consider, for example, blading of 100 mm height, for 
which a pitch of 20 mm at the mean diameter is not abnormally small. If the sine of the 
exit angle is 0.5, the average width of the layer is 0.3 multiplied by 20 mm, or 6 mm, 
so that roughly it may be said that the layer has a rectangular crc ee-section of about 
100 mm height and 6 mm width. This indicates that for the problems ^ -nsidered in this 
chapter the assumption of infinitely small pitch is satisfactory. 


102 


2321 















6.4. Three-Dimensional Potential Flow 
with Inclined Stationary Blades 


(a) Statement of Problem . On the upstream side of a stationary "blade grid 
there is potential flow. The circulation 


I" 7 0 " 2 71 r c u0 


( 1 ) 


which is independent of r, is given. The "blade shape is such as to result in a certain 
variation of exit angle ^ ^ with the radius, the functional relationship "being ex¬ 
pressed "by 


tg ct i = fal' = f( r ) 

c ul , 


( 2 ) 


The volume flow Q is also given. The meridian passage is "bounded "by cylindrical 

walls. 

The variation along the radius of the angle of Inclination l? is to "be found. 

This is the angle which the stationary "blade, considered as a vortex filament, must make 
with the radial direction if the flow on the downstream side is to be irrotational (see 
Figure 25). The meridian stream line pattern is also to be determined. 

It is apparent that for the solution of this problem additional data are nec¬ 
essary. At the further end of the region on the downstream side of the grid, same bound¬ 
ary condition is needed, and this is furnished by the next row of blades, which will of 
course influence the meridian flow. Naturally the conditions thus imposed must be 
introduced into the calculation. An analogous condition exists in the region on the up¬ 
stream side. In order to keep the details of the calculation within reasonable bounds, 
consideration will be limited to the two following cases, both of which are of importance 
from the practical standpoint. 

Case 1. Given only a single row of blades affecting the meridian flow (see 
Figure 21). 

Case 2. Given an indefinite number of identical, uniformly spaced rows of 
stationary blades affecting the meridian flow, as shown in Figure 26. Since between any 
two such rows the flow is the same, it is sufficient to consider only one section made up 
of two rows. 

Case 2 therefore corresponds to a homogeneous group of stages. It is considered 
that only the stationary grids are the "rows of blades affecting the meridian flow" 
since for mechanical reasons the moving blades can hardly be inclined at an angle to the 
radial direction. moving blades should therefore be so shaped that for the given meridian 
stream line pattern, the circulation along each blade is constant; otherwise the potential 
flow would be distorted. If treatment of a more general case is of interest, the analysis 
will In general be similar; no fundamentally new conceptions of any kind would have to 
be introduced. 

(b) Basic Equations. All lengths are divided by the blade inner radius (hub 
radius) r to obtain dimensionless expressions, and dimensionless "characteristic" 
velocities are introduced by dividing the actual velocities by the rotor velocity at 
this radius. With the dimensionless cylindrical coordinates x, y, and 17" (see 
Figure 22) are associated the unit vectors i, and k, respectively. The characteristic 
velocity C is in general 


C r 


c 


c 


( 1 ) 


”n 



105 


2321 






Let 


C 




( 2 ) 


c a y ay 


(5) 


so that the total volume flow Q* (also dimensionless) is 

Y 

Q* : 2 TT 

1 

It is found desirable in the calculation to make use of the fact that for potential 
flow the law of_superposition is valid. The velocity C is therefore resolved into a 
"basic velooity C and a superposed velocity as follows. Let 


Y 

/ 


Then 


Q* r 0 * 

n. <jr (i 2 -1) 


C s Cgi^ -f C-yk 

1 = Qal + <l r l 

- = I +■ SL 


00 

(5) 

( 6 ) 
(7) 


£ is therefore the additional flow which must_be superposed on the "basic through flow 
to obtain the actual three-dimensional flow. C has no radial component; £ has no 
tangential component. 

At the exit of the stationary blade grid. Equation 6,k (2) is valid, so that 


c al 

c ul 

i tg 

<x 1 = 

f(y) 

(8) 

Since there is to be potential flow 

on the discharge side 

of the stationary blades. 

C ulY 

Eram. (8) and (9) 

- const. 

. Hi 

zir 


(9) 

Cal^ 

= c uiy 

f(y) : 

P* 

-h- e (y) 
z ir 

(10) 

If this equation is integrated from 1 to Y, the left-hand 
can be seen immediately from (3). Therefore 

side becomes 

Q*/2 7T , 



L 



0 * = 

rl 

A f(y) 

ay 


* 

: 


o* 




Y 

/ 

f(y) dy 



- 

1 



(H) 


10k 


2321 








can be 


In this equation the right-hand side is completely known, so that * 

calculated. From (10), C a ]_ may then be expressed as a function of y; i.e., 


'al 


-L_ f(y) 


2 nr y 

Equation (7) is now written for the axial component. 


( 12 ) 


'al 


+ <lal 


so that finally 


al 


'al 


al = 




b. f (y) r 


Q* 


'H' (y 2 -iy 


(13) 


™ = g(y) (14) 


The basic flow is now completely determined. On the upstream side of the stationary 
row considered, 


°0 = + °U0*1 ■ _£_ 1 + r* i (15) 

TT (I 2 -1) 2 T y 


and on the downstream side 


£ = °ai + = --2^— i + JjL A ( 16 ) 

7T (3T -1) 2 -7T y 


In Equations (l4), and (15), and (l6) everything on the right-hand side is assumed to 
be known except *, which is calculated from (11). It Is considered t&at q^ 

has the value given by (1^) only Just as the exit of the stationary grid (mathematically 
speaking, in the plane of the grid), while C uQ and are constant values for the 

entire region between the stationary row of blades considered and the preceding or the 
following moving row. From continuity considerations, there can be no change of q a in 
the plane of the stationary row, so that q^ = q al . 


For the superposed flow the boundary condition (l4) at the plane of the stationary 
grid is known. At the walls q has no normal component, so that 


^rC 1 ) 


0 ; 


<l r (Y) = 0 


(IT) 


If §> is the potential of the superposed flow, the problem may now be stated as 
follows. The equation 


± - o 


a* 2 - &t 

is to be solved with the boundary conditions 

d$ 

c><J 


(18) 


Qr(l) 


3a0 


1- 

%1 


<1 (Y) 

dx 


dl I 

: g(y) 


(19) 

( 20 ) 


105 


2321 














at the plane of the stationary grid. This is a problem of the "classical" type. As a 
solution let 


= F(x)G(y) 

so that Equation (18) becomes 


( 21 ) 

9 . 


GF" + FG" FG* : 0 (22) 

7 

where the prime (’) and second (") indicate first and second derivatives, respectively, 
with respect to x or y. (Confusion is avoided by reference to (21).) Equation (22) can 
also be written 


F" 

~F 


G" , G* 

G yG 


(23) 


The expression in F, as well as that in G, must be equal to a constant (called lc^), 
since otherwise (21) could not in general be satisfied (because F depends only on x, 
and G only on y). It is evident that (22) is separable into two differential equations; 
namely, into 


with the general solution 


F" 



F 


F 


k]e ■ 3C -f 



(24) 

(25) 


and into 


with the general solution 

G r ZqOqO 


G" 


+ 


2 

1 G' + KG 

y 


0 



cos ft +- N 0 (iy) 



( 26 ) 

(27) 


Jq is the Bessel function and Nq the Neumann cylindrical function of zero order. For 
brevity, their linear combination (constants A and*'/3 ) will be denoted by Zq. It is now 
merely necessary to construct systems of functions from expressions of the form ( 25 ) or 
( 27 ) which satisfy the given boundary conditions. 


In accordance with (21), for the two equations (19) there can be substituted 


G’(l) = 0, 


G'(Y) 


Since it is shown in the theory of cylindrical functions that Zq* 
(28 ) may in view of ( 27 ), be written 


Jl(k) cos ft -f- Np (k) 

J x (kY) go* ft + N x (kY) 

This system of equations is equivalent to 

J 1 (kY)N 1 (k) - J 1 (k)N 1 (kY) : 0 

Jp(k) 






IMk) 


ein ft 
sin ft 


( 28 ) 


Z]_, Equation 


(29) 

(30) 


106 


2321 






Equation (29) is a transcendental equation for the determination of k. It has an infinite 

number of roots k]_, k2, k^ . .., kj. . ....... which are all positive, 

and -which are called the characteristic values o f the problem. 

The numerical evaluation of the roots is carried out in such a way that each time 
+ 1 ^ , with each characteristic value k^ there is associated a value of / /$ i 

given by (30). Equation (27) also gives a system of functions ZQ(k i y), Z^kgy) .... 
Z 0 (kjy) .... (except for a certain constant A which is not important in the present 

problem) which are the characteristic functions of the problem. The characteristic values 
may be determined - that is, equation (29) may be solved - either graphically or by a 
process of approach (trial and error). The values of the functions of and are 

taken from published tables (for example, Jahhke and Emde). For large values of the 
argument which may not be included in the tables, approximate formulas given in literature 
pertaining to this subject may be used. It still remains to satisfy the boundary con¬ 
dition (20); the procedure required will be different for Cases 1 and 2. 


(c) Case 1: Single Stationary Grid, Figure 21 . Since in this case the grid plane 
is located at x = 0, Equation (20) becomes' 


_2f I - g(y) ' (31 

D* [ 

X=0 


The "other end" of the region considered is at x - oo • There, evidently, q - 0, since 

the superposed flow must be such that if it alone were present, the total volume flow 
between the two boundary walls of the meridian passage would be zero. As a further 
condition, then. 


From (21) and (25) 


af I 
"5T 1 


C# 


0 


(32) 


% 


af 

a* 

/ 

so that since G 


% 


0. 


GF* 

+ 


= Gk(K 2 e 
and k ^ 


“ ^ e ~ laC ) (33) 

0, condition (32) can be satisfied only with 


In order to satisfy (31), it is necessary to substitute for G the general linear 
combination of the functions, Zo^y), Vhich naturally satisfies the differential 
Equation (26) and the conditions (28) Just as well as each of the functions taken alone 
would satisfy them. Instead of (33), therefore, a complete expression would be 


_ °° 

- < 
2>x i = l 



Zo^y) 


m 


In the coefficients A* are included the A and Kg previously used. Condition■ (31) then 
finally becomes 

OO 

g(y) = AiZo(kjy) (35) 

* 


107 


2321 






It Is therefor© necessary to replace the given function g(y) by a series expansion of 
ZQ^y), a problem which is completely analogous to that of a Fourier analysis. In the 

theory of cylindrical functions it Is shown that if the values of k^ are determined "by 
an equation of the type of (29), the system of functions 'i|'yZ 0 (k i y) is orthogonal in 
the interval between 1 and Y. The formula fof the coefficients is 


A i 


1 

-A i 



7 g(y) Zq (*iy)ay 


with 

Ai= ^ y [ z 0 (k ± y) J 



1 

2 


A 


0 2 (k 1 Y)-Z c 



(36) 


(37) 


In evaluating the integral it has been assumed that Z = 0 at y = 1 and y - Y, 

In accordance with the boundary conditions ( 28 ). 


The process of calculation may be summarized aB follows. If the function g(y) has 
been determined from (l4), the characteristic values k-^, k^, . . • k^ • . . are calcu¬ 
lated from (29), and the values of /& %.* . /3 • from (30). Then 

Zo (kty) c cos i J 0 (k 1 y) + sin /3 ^(k^y) (38) 

Z 1 (kj^y) = cos p iJ 1 (k i y) -f sin ^ ^(k^y) (39) 

In these equations the values of yii are obtained from ( 37 ), and the values of 
from ( 36 ). The final result may be written 


- - 

<X> 

£ n 

i - 1 

%. = 

2 
i - 1 

00 

«r = 

$ 
i = 1 

is given 

by the 

total flow (not 


-kjx 


Zo(lCiY) 


Ai© klX ^(^y) 


Aie Zi (kj_y) 


( 8 ), using dimensionless characteristic velocities and lengths, 


m 




m 


? ■ / 


yO a dy = 


oO 


4- + £ £ 

k i 


Y(0 a +■ qa) &y z 
e ^*7 Z 1 (k i y) 


i = 1 


PS) 


108 


2321 









The arbitrary integration constant is made equal to zero. The stream lines are lines of 
constant W . It is to be noted that all these equations are valid for x 0. 

Since the stream line pattern is symmetrical about the plane of the grid. 


§ (-x,y) 

ia (-*>y) 

(-x,y) 
If (-x,y) 


- $ (x,y) 
t qa(x,y) 


■ ^r(x,y) 

+ lf(x,y) 


m 


(d) Case 2. Similar Stationary Grids Uniformly Spaced, Figure 26 . Consider the 
space between two stationary grid planes, one of which is located at x - ~ a/2, the other 
x = t a/2 . Since in both planes qa is equal to the same function g(y), it is evident 
that 



z><£ 






(- 


t) = F'6 


(X 


(*5) 


or, from (25) 


k x e 





-ka 

- K2e 2 ,* 


*1 = Kg 


(46) 


The eaponential expression F becomes in this case a hyperbolic function. The rest 
of the analysis is exactly the same as before, and the process of calculation is similar 
in every detail to that bf Case 1. The final result only is given 


$■ 


<la = 


oO 

Ai 

- 1 k-jCh/ a k^ 

OO 

2 

A i 

i -1 Ch 

(**l) 

oO 

. A i 

i=lCh 

fa k \ 

\2 lJ 

OO 


Sh(k 1 x)Zo(k 1 y) 


Ch(k i x)Z 0 (k i y) 


Sh(k 1 x)Z 1 (lc 1 y) 


Hr = +. 2 


Ch(k 1 x)yZ 1 (lc L y) 


i = 1 kiCh^a k^ 

These equations are valid in the interval ~ ^/z = * = ^ . lJ/~ 


(47) 


(48) 


(49) 


(50) 


is again the 


stream function of the total flow, and not merely of the superposed flow. The term 
Ch(a/2 k.) in the denominator of the coefficients of these equations might be included 
in Ai itself; it was introduced here so that the coefficient formula (36) would remain 
unchanged. 


109 


2321 










(e) Angle of Inclination. It is evident that the tangent of the angle of 
inclination // is equal to the ratio of the radial to the tangential velocity in the 
plane of the stationary grid. That is, 


tg ^ 


C rl~ C rO 

C ul“ G uO 


Q-rl-3-rO 

C ul" C uO 


.ZJljL _ (4rl-4r0 > (51) 

Tl *" >0 * 


For Case 1, "because of the symmetry of the streamline pattern, this equation becomes 


tg \> = 2q rl 

C ul " C u0 


k ^ Jq rl 



(52) 


Equation (52) is also valid for Case 2 (that is, for a homogeneous group of stages) with 
the exception of the first and last stationary blade grids. 

6.5. Numerical Example for Section 6.4. 

There is to be calculated the streamline pattern and the variation of along y 
for an axial compressor stage with radius ratio Y = 1.4 and axial length a r 0.4. 

The two "bounding" rows of stationary blades at x - -a/2 - -0.2 and x = +- a/2 

* -f 0.2 are identical in construction (case 2). The stage belongs to a homogeneous 
group. Let the inner diameter be the principal diameter, and let ^ s a 0.5; 

Tig r 0*^5; = 0.7. At y : 1 (inner diameter), tan 1 » 0.75, and 

the variation of d- 1 1 with the radius is given by 


tg*! . f<7> = tE 0 (k l y) + °al . (« 

C ul 

where M is a constant. It will shortly be evident that the function f(y) is completely 
determined. With this form of function, the problem is particularly simple, because 
Zo(k-j_y) is the first characteristic function of the problem. The "series development" of 

g(y) then consists merely of a single term, the coefficient A]_ of which is obtained 
directly without the necessity of an integration of 6.4 ( 36 ), which in general would have 

to be made graphically. 

It is evident that since the inner diameter has been selected as the principal dia¬ 
meter, and since the characteristic velocity G is referred to the rotor velocity at the 
inner diameter, for the case of potential flow here assumed. 


W 1 ) - los - ^23 

Cul(l) = T 1S 

: f S 


Therefore the given data may be written as follows; 


110 


2321 

















c u 0 = 

0.70/y 

Y = 1.4 


c = 

0.45/y 

a - 0.4 


c - 

0.30 



f(y) - y MZo^iy) 

+■ 0.30 

0.45 

with f(l) = 0.75 

( 2 ) (3) 

In view of 6.4 ( 9 ), Equation 6.4 (l4) becomes 



*al = 

g(y) * mz q 

( k iy) 

(4) 

Solution of 6.4 (29) and ( 30 ) with Y 

= 1.4 results in the values 


k l 

= 7.89, 



cos - 

0.644, 



sin^a = 

0.765, 



so that 




Z 0 ( k ! 7 ) = 0.644 J Q ( 7 . 897 ) 

•h 0.765 

N 0 (7.897) 

(5) 


In particular, for y = 1, Z Q : 0.2833, so that from (2) and (3) 

f(l) = 0.75 = 0.2833 * M t 0.30 

Therefore M s 0.1325. The complete solution may now "be written. Equation 6.4 (48) 
for this case becomes 

<3al = _ h _Ch a k, Z n (k,y) 

Cha k x 2 01 

2 

so that, as shown by comparison with (4), A± = M. Therefore, with Ch(a/2 kj) = Ch(0.2 
7.89) = 2.525 

q a : + 0.0525 Ch (7.89 x) Zq (7.097) ( 6 ) 

q r - “ 0.0525 Sh (7.89 x) Z 1 ( 7 . 897 ) (7) 

= 0.15 y 2 +• O.OO 665 Ch (7.89x)yZ!(7.89y) ( 8 ) 

tg s z - 0.973 7 Z]_ (7.897) (9) 

Z is to be calculated from (5), and Zj_ is correspondingly found, using J]_ and . The 

sign of tan iS merely indicates on which side the blades are inclined, and since this can 
always be determined immediately by inspection, the sign is not particularly important. 

> 

The calculated results are given in Figure 27, which shows the meridian streamline 
pattern, the variation of the axial component C a along y for the stationary and moving grid 
planes, the change in direction of the "stationary grid axis" (that is, of the vortex fila¬ 
ment which is substituted for the stationary blade) as shown by the variation of »>, and the 
variation offlCi. There is shown also a plot of norm by which is to be understood 


111 


2321 






the angle c\ which would exist in the corresponding normal stage with all stationary 
blades perfectly radial. The following conclusions may be drawn. 

To a very marked inclination of the stationary blades there will correspond a 
noticeable (though not exceptionally large) effect on the distribution of the axial 
velocity. The meridian streamline pattern depends directly upon the distribution of C a 
(or more accurately expressed, upon the change of this distribution along x), but the 
effect is not particularly important. On the other hand, the meridian streamline pattern 
affects the variation of C a very appreciably. Equation (51) shows, moreover, that in 

the case of turbines an inclination of the stationary blades influences the variation of 
C a and the meridian streamline pattern much more than in the case of compressors. Be¬ 
cause of the greater deflection, -ri * - ro* is much larger, so that to a given 
tangent p there corresponds a very much greater cross-flow velocity q r 2 _. 

In conclusion, the following remarks may bo made. It is a very great simplification 
to replace rows of blades by plane unstable surfaces. This substitution involves the 
least approximation if the "blade plane" is so located that it passes through the "center 
of gravity of the axial distribution of circulation" of the row of blades (it should bo 
clear withovit further explanation just what is meant by this). The actual streamline 
pattern lias naturally no instabilities, and is obtained from the theoretical pattern by 
"smoothing out". It is to be noted, however, that the theoretical streamline pattern at 
the place of most practical interest - that is, at the exit - already conforms very closely 
to the actual pattern. It is evident that the tfC ^ and C a ]_ of the numerical example 

are not values Which actually occur. A better approximation for the actual case would 
be obtained if the values in question were taken not at x = -a/2, but at the actual 

exit of the stationary blade. 

It is conceivable that the theory night be extended by replacing the rear of blo.des 
not only by one, but by several successive unstable surfaces, or even by substituting 
eventually an infinite number of such surfaces (that is, a continuous vortex region). 

The calculation would then be correspondingly ‘more complicated, and such a refinement 
would have very little significance, since such studies are ordinarily intended to give 
only the principal data and to estimate orders of magnitude, for this reason a very 
special variation of tg (A ^ - -f(y) was assumed in the numerical example, although it 

could clearly be foreseen that the result would be an S-shaped blade axis which it would 
hardly be practicable to use. However, this is of no great importance, since there would 
be no object in constructing such blades to satisfy with great accuracy the requirements 
of the potential theory. 


The question of the inclination of the stationary blades is particularly important 
in the case of axial compressors, where the reduction of noise may be an important nroblen. 
E'.U’then.iore, it is possible by such means to affect to a certain extent the angle of flow, 
and so under certain conditions to improve the grid properties at one of the mo3t critical 
places (that is, at the inner diameter). finally, an engineering simplification is also 
achieved if it is possible to use stationary blades which ere not warped. 


6.6. Rotational Flow through the 
Ilonogeneous Stage 


Consider the flow of a frictionless fluid tlirough a homogeneous stage, the stationary 
and moving blades of which are both radial, and along the radius of which there is a 
specified variation of the exit angle , or /3 2 . In general, a potential flow will 
not be possible, and the problem presented is: what will be the nature of the flow if 
the blades are constructed in a way which does not correspond to the requirements of the 
potential theory? This problem lias already been discussed quite thoroughly under 6 . 3 , 
where it was ascertained that in the middle of a group of identical stages the flow 
process periodically repeats itself. It was also pointed out, in particular, that the 


112 


2321 


circulation (calculated for paths lying on stream surfaces) is constant along every "blade, 
in spite of the fact that rot £ ^0, so that the energy conversion is the same for every 
stream surface. One period ox this periodic process is considered here. 

For the extraordinary complexity of the problem, it is clear that a mathematically 
exact solution cannot "be obtained, and that material simplifications must be made. Each 
of the three velocity components is a periodic function of x (see Figure 28 for the co¬ 
ordinate system), and so can be expressed by a Fourier series. The coefficients of 
this series are functions of y. As a first simplification, only the first order of this 
Fourier series vlll be retained. In theory, it is possible to introduce the simplified 
expressions into the Euler equations of motion, ancl thus to obtain differential equations 
for the Fourier coefficients as functions of y. Practically, however, the solution of 
such a system of simultaneous differential equations is impossible, not merely because of 
their complexity, but particularly because the introduction of the conditions which are 
imposed upon the flow by the blading would give rise to mathematically insurmountable 
difficulties. 


The second simplification consists of abandoning one integration of the general 
equations of motion, and of satisfying the equations in only two specified planes. 

These reference planes are located in the axial space between the rows of blades, ac 
shown in Figure 28. It is further assumed that the axial component of the velocity in 
the two reference planes (denoted by the subscripts 1 and 2) varies linearily with the 
radius. With an arbitrarily specified function 

ctg 1 - f(y) ; ctg yS 2 - g(y) (1) 

it is no longer possible to specify that the circulation along the bladec shall be 
exactly constant. It is found, however, that for the functions x(y) and g(y) which 
are of practical importance, this condition can actually be very closely maintained. 

The condition will here be expressed for the only two stream surfaces known in advance; 
namely the cylindrical bounding walls of the meridian passage. The process of solution 
based on these assumptions is given in the following paragraphs. 


The functions f and g of Equation (l) are given, as well as the worlc coefficient 
\ s of the stage referred to the inner diameter as principal stream surface. The 

same characteristic velocities are introduced as were vised in Section 6.4. At the inner 
and outer diameters (i.e., at y = 1 and y = Y) 


An 

: ®pn - 2 (Cul “ 

c u2 ) 

(2) 

X G 

11 

ro 

CO 

11 

ro 

CO 

& 

1 

c u2 

(3) 


y 2 

The condition for equal circulation at y : 1 and y = Y is that An = ^ a Y^ 

It is further to be noted that in general 

ctg ! : f ( y ) ( 4 

^al 


ctg 2 


*u2 _ 7 ~ C u2 

c a2 C a2 


g(y) 


(5) 


as can be derived immediately from the velocity triangles. If C ul and are calcu¬ 

lated from (4) and (5) and substituted in (2) and (3), letting y = 1 and y : Y, 


113 


2321 




respectively, and if- X n and \ B a^e replaced by their equivalents in terms of 
X s in accordance with the specified condition, there is obtained finally 


X, 


2 [c al (l)f(l) -h C a2 (l)g(l) 

- 

( 6 ) 

2 Y [c a i(Y)f(Y) 4- C a 2 d)g(Y) 

- r] 

( 7 ) 

■ are now substituted for S a q and 

c a2 . By 


c a i = A i + B i y 


( 0 ) 

c a2 = *2 -f D 2 7 


( 9 ) 

s 1 and y a Y, the C a q(l), 

Ca 2 (D^ C al (Y), 

and C, 


occurring in (6) and (7) can be expressed in terms of the coefficients Aq, Bq, Ag, and Bg- 

Furthermore, an expression may be obtained for the value of Stokes' stream function at 
y = Y, if it is taken as zero at y : 1, 

Y * 


(Y) 


yCal dy r Aq Y - 1 , Bq 

2 3 


Y 3 -1 (10) 


Also, however, 


f (Y) 


yC a2 dy : Ag Y 5 - 1 + B 2 Y 3 -1 (11) 


Therefore, since y = Y is a streamline, and (Y) consequently cannot depend upon 

x, it follows that 


At Y 2 -1 
2 


4- 


B 1 y3 - 1 = ^2 Y 2 -] 


4- 2 


Bo Y"* -1 


( 12 ) 


As is evident from (8) and (9), Equations (6), (7), and (12) constitute a system of 
equations which may be expressed as follows. 


Al ^ -1 . Bq Y 3 -1 

2 ' 3 


A 2 


Y 2 


Aqf(l) +- Bqf(l) 4 _ Aged) +' Bged) 
Aqf (Y) -f BqYf(Y) + Ag g (Y) 4- BgYgfr) 


B2 Y 5 -1 


% 


2 

X 


2Y 


( 13 ) 


In this system of Equations Aq, Bq, Ag, and Bg are unknown. If one of these coefficients 
is arbitrarily given a certain value, the other three are determined by the system. It 
is best to select Bq as the arbitrary coefficient (at least for the time being), chiefly 
because of its physical significance; i.e., Bq is a measure of the amount of change of 

the axial component along y at the exit of the stationary blades. Tire system of 
Equation (13) may then be solved as follows. 
















A 1 I i 


Ag s 1 
D 


B 2 = 


1 

D 


where 


D = 


-*1 


Y 3 -l 


Y 2 -! 

2 


_ Y 3 -l 


[ 


+ 1 - B l f (!) j 

g(l) 

g(l) 

-ST** -B iy f(T) ] 

g(Y) 

Yg(Y) 

Y^-1 Y 3 -l 

— - B l — 


- Y 3 -! 

3 

f (!) [J^B t 1 - 

B x f(l) 

j g(D 

f(Y) f Y-BjYf (Y) j 

Yg(Y) 


(14) 


(15) 


yg-i 

2 

f(l) 

f(Y) 

Y 2 -! 

2 

Y(l) 

f(Y) 


- Y2-1 

2 

6 ( 1 ) 

S(Y) 

- Y 2 -] 


-B x Y5-1 


[^8 + l-Bjfd)] 
[~ + Y -BlYf(Y)] 


(16) 


8 ( 1 ) 

g(Y) 


- Y 3 -! 


g(l) 

Yg(Y) 


(17) 


It is now possible to select different trial values of B^ and to determine the correspond¬ 
ing values of A 1# Ag, and Bg by use of Equations (l4) to (17). Each of these trial 

solutions satisfies the continuity equation and the circulation condition. The latter 
is, of co^^rse, only completely satisfied at y r 1 and y s Y, but as has previously 
been mentioned, it is satisfied approximately at intermediate points. The problem is 
then merely that of finding the value of B]_ which best satisfies the general equations 
of motion in the plane of reference. 

Stokes' stream function is now introduced again, and the two following equations, 
which are analogous to (10) and (ll), may in general be written for the two reference 
planes. 


A 1 + B 1 £=1 
2 3 


(18) 

(19) 


\jf i(y) = * y) 

^ (y) zW (a,y) = A 2 y 2 -l 4 Bg y5-l 
* 2 3 

If it is now assumed that ijr depends upon, x in accordance with an equation of the form 
Ip* (x,y) = ¥o(y)+A$(y) COS 2TTx (20) 


115 


2321 


















it is evident that (18), (19), and (20) can be simultaneously valid only if 


, A$ s -Vi 

2 -o- 


+ Hf 2 fo) - [t! r 1 (y) - 2 (y)j cos 2^x 1 ( 21 ) 


Use of Equation (20) tacitly assumes that only the first order of the Fourier 
series through vhich C a and Cx* may be expressed is to'be retained. There is, how¬ 
ever, a further assumption included in this equation which concerns the phase of this 
first, order. According to (20), those points of the meridian streamline which have 
tangents parallel to the x-axis are at- x = 0, x = a/2, x = a; that is, at the middle of 

the axial spaces between the blade grids; or otherwise expressed, the cross-flows in the 
axial clearance spaces reverse direction at these places. In effect, it is assumed that 
the distinguishing features of the flow are to be found in the axial spaces between the 
blade grids (in which the flow "is left to itself"); that is, that either the "peaks" 
and "depressions" or the points of inflection of the wave-shaped meridian streamlines 
occur there. The latter, however, is forbidden by (l8) and (19), so that the peaks 
and depressions must occur at these places, and this fact is expressed by Equation (20). 

/ 

In the limiting case of a very great axial interspace between the blade grids, 
the assumption may be considered to be correct, since in this case there is flow only in 
a very long space bounded by. cylindrical surfaces, and the flow surfaces are practically 
cylindrical except at the two ends. It cannot be denied, however, that the theory has 
a weakness at this point, since the assumption contained in (20) can be justified only 
by its plausibility. 


¥< 


Therefore 


¥ = i 


r 


^(y) 


The general equation of motion 6.3 (3) is now introduced. For brevity let 


E 





so that E is a dimensionless number representing the total energy. If F 
dimensionless expression of 6.3 (3) is 


( 22 ) 

0 , the 


C x rot C = grad E 


(23) 


In the axial interspace between the blade grids, F - 0. Furthermore, E is everywhere 

constant on a stream surface except in the moving grid itself, so that the vector grad E 
is perpendicular to the stream surfaces. Since the planes tangent to the stream surfaces 
are parallel to the x-axis in the planes of reference (l) and (2),‘ grad E is radial in 
these reference planes. It is therefore merely necessary to use the proper component 
equation for the radial component from (23), or 


However, 


Eh JL (y 0u) 

y l>y 



(25) 

Sy 1 ^STj 


= 3e P? 


116 


2321 















and from Equation' 6.3 (8) defining the Stokes* stream function, 


so that 


JL2L - 7 C, 

D<J 


c)E 

ay 


Equation (25) therefore becomes 


C 


u 




2 


^ (y 
<^y 


Cu) “ 


r y c a 3 E 

1 f ^ Cy - 3 °a 
y v ^ * dy 



(26) 


Since by assumption the energy conversion per unit mass is the same for every stream 
surface, the total change A E occurring in the moving grid is independent of , 

■while everywhere except in the moving grid E depends primarily upon ij/' alone. It 
follows that <)E/^ depends only upon ijT everywhere in the stage. This gives 

a criterion which can be used in both reference planes. For a physically possible flow 
rigorously satisfying the equations of motion, the following relation must hold for any 
specified value of IF • 


E 1 

<3 E 

df 


1 

s a 

X n a' 


2 


(27) 


Because of the simplifications which have been introduced, it is not possible to find 
a solution which exactly satisfies this equation for all possible values of All 

that can be done is to find the best "average" solution, and this is accomplished by 
requiring that the integral 



(28) 


be a minimum. The integral is to be evaluated with the help of (26), the entire left- • 
hand side of which can be calculated. C a is given by (8) or (9)> so that 


Furthermore, 





= Ss (29) (30) 
dy 


G ul = C al f M> 


* 

c) y 


(y C u ) 


X 



[yc al f(y)] 


(31) (32) 


c u2 = y- c a2S(y )> 


<^y 


X 



{^"y^sCy)] (33) (3*0 


117 


2321 























For calculation of the left-hand side of (26) there is now lacking only S C r / S ~r_ f 
This may he obtained with the help of (21). 


Gr = " I 

y 


a x 


so that, from (21), 


<) C. 


r. 


x 


2 t r 


In particular, at the reference planes, 


^G r 

Dx 


21T 

2 

a y 


ii=.i 

^ x y 


a, 2 


[ ¥ i (y) 


- ? 8 w) 


^i (y) - -f a (y) 


cos 2 IT x (35) 
a 


(36) 


x z a 
2 

^G^ 


5 x 


- - 2 TT 


x=a 


a 2 y 


7 Y x (y) "^2^1 (37) 


The right-hand side of (36) can now be completely calculated; that is, ^ E /^$ r is 
determined. Although all quantities are given as functions of y rather than of 1 ifr 
Equations (l8) and (19) at x ■ a/2 and x = a permit the determination of the value of 
¥ corresponding to every value of y, so that it is finally possible to obtain 
d E/j as a function of ^ 


The process of calculation may be summarized as follows. The angle functions f(y) 
and g(y), the work coefficient X 3* and the characteristic dimensionless lengths Y 
and "a" are given. The first step is to select an arbitrary value of B-j_ and to calculate 


A^, A 2, and from Equations (l4) to (17). C &1 and C &2 are then obtained from (8) and 

(9). Using (29) to (37), the entire left-hand side of (26) is computed, first for x - a/2, 


and then for x r a. is thus obtained as a function of y, and using 

(18) and (19) it is then expressed as a function of ~\$r . The latter functional relation¬ 
ship is plotted, and the difference between the curves for x a a/2 and x a a is taken. 


In accordance with (28), this difference is squared. From a graphical integration the 
value of the integral expression F 0 corresponding to the selected Bp is obtained. 


This process is carried through for several values of B-j_ > and a curve is plotted showing 


F Q as a function of B-j_. The value of B]_ at the minimum point of the curve is taken as 
correct - that is, as correspondingly most nearly to the actual flow conditions. 


It is conceivable that the method might be perfected by introducing into (20) a 
variable shift of phase, or by using a more general expression for C a . The calculation 

would then become very complicated, however - much more complicated than if the simple 
method were first used to obtain an approximately correct -solution which could be improved, 
if desired, by further refinements. To obtain a higher degree of accuracy, it would be 
necessary to use more reference planes, especially in close proximity to the blade grid, 
but this would introduce very great complications. 


118 


2321 
















6.7* Numerical Example for Section 6.6 

The flow through a usual reaction turbine stage has been calculated in accordance 
with the method described. The following data were given. 


A s 

= 

3.2 

(referred to 

Y 

= 

1.4 


a 

= 

0.3 


ctg* L 

= 

f(Y)' 

• = 10.468 

ot 8/? 2 

as 

g(y) 

3.187 

the moving 

blades 

are assumed 


9.940 y + 


be identical. The variation of angle 
is so selected that it corresponds approximately to an "unvarped" blade. This does not 
mean that the exit angle is constant along the radius, as might at first be supposed. 

The blade grid has the same profile at different radii, but a different pitch, and this 
results in very different deflection properties. The variation of the exit angle nay 
often be approximated by the square law, as in the present case. 


Carrying through the process of calculation as described in 6.6, the curve shown 


In Figure 30 is 
occurs at B^ = 

obtained. This gives F 0 
0.130, where 

as 

a function 

of Bi . The Tni-ninnim value 


F o 

■= 

0.0795 




Cal 

- 

0.2477 


O.UOy 


C a2 

= . 

-0.0697 

+- 

0.392y 


Figure 29 shows d- ^(y) = /& 2 ( 7 ) > c al > and> c a2 as functions of y, and also shows 

the meridian streamline pattern. 


Figure 31 gives the velocity triangles for the different stream surfaces, and 
Figure 32 shows the degree of reaction Q for the different surfaces as a function of 
y-^ (i.e., it shows Q for those flow surfaces which at x = a/2 pass through y - y^). 

Figure 32 also shows the variation of the degree of reaction & * which a normal stage 
(with "correctly" shaped blades) would have with 50$ reaction at the mean diameter, and 
shows the line 0 o z constant ■ 0.5> which corresponds to the simplest theory 

(constant degree of reaction along the blade). The chief conclusions to be drawn are 
as follows. 

The axial velocities are greater at the outer diameter than at the inner, so that 
the flow, on the whole, is forced toward the outside. Therefore cross-flows arise, but 
these are not of great magnitude. Near the stationary blades the flow is inward, and in 
the moving blades it is outward. The meridian streamlines are predominantly "straight", 
so that the velocity triangles for the mean stream surface are very accurately those 
assumed in the elementary theory. The degree of reaction changes considerably along the 
blade; it is approximately a mean between the constant value assumed in the elementary 
theory and the value for a normal stage. 

The accuracy of the results can be proved only by test. However, it is encouraging 
to note that the value of the integral* Fo is ten times greater for Bi ■ 0.04 and Bi ■ 

0.22 than for the value B]_ s 0.13 which was taken as correct. If the circulation 
condition is checked for the middle stream surface (in the calculation itself the 
circulation was considered only at y = 1 and y ■ Y) it is found that an error of only 
about 1$ was introduced. 


119 


2321 




In conclusion, a theoretical question of some interest will "be "briefly considered. 
"What is the loss caused "by the use of "blades which are not properly shaped? For friction- 
less fluids this loss is caused simply by the fact that there can be no potential flow 
on the downstream side of the last stage, and consequently the exit loss is somewhat 
greater. This follows from the fact that the kinetic energy content of a rotational 
flow is greater than that of a potential flow with the same normal components at the 
bounding surfaces of the region considered (Kelvin’s Law). However, the exit loss from 
multi-stage machines is in itself relatively small, and even a very considerable in¬ 
crease (such as is considered here) will have such a small effect on the overall efficiency 
of the machine that in most cases it may be neglected. 

A noticeable overall loss can usually arise only as a result of the effect of the 
viscosity of the fluid. For a rotational flow with viscous fluid, there is a dissipation 
of energy voider the conditions which exist in a turbo-machine, the greater part of which 
is not ascribable to the direct effects of the viscosity itself, but to the effect of the 
turbulence which is present. However, this loss cannot be very great, since the strengths 
of the vortices (i.e., eddies) of the main stream are relatively small - in fact, very 
much smaller than the vortex strengths in the boundary layers where the profile and wall 
losses originate. For a given (constant) viscosity the energy dissipated is proportional 
to the square of the vortex strength. For a viscosity which is itself dependent on the 
local stream conditions, as is the case when there is turbulence, the energy dissipated 
depends upon a power greater than the square. It is therefore to be expected that such 
dissipation losses will be of a far lower order of magnitude than the other losses. 

An i'tportant overall loss can occur only by reason of the separation of flow likely 
to occur with blades which are not correctly shaped, as is the case for "unwarped" blades 
over a considerable portion of their length. This may be one reason why the Ljungstrom 
blading used in axial turbines, which is very insensitive to the angle of flow, has been 
found to be just afc good as other blading which inherently should have better grid proper¬ 
ties with the proper flow conditions. 

From the foregoing analysis it appears that a blading designed in accordance with 
the principles of the potential theory is not necessarily the best. If the blading is 
not designed in accordance with this theory, but is nevertheless such that separation 
at the blade surface in consequence of unfavorable flow conditions is avoided - a design 
for which the theory given under 6.6 affords a basis of calculation - then it is not to 
be expected that important additional losses will occur. It Is even possible that a 
better efficiency may be obtained, because the clearance losses may be smaller with a 
different variation of the degree of reaction along the blade. It is noteworthy that 
the induced losses in a multistage blading are inherently very small, and relatively 
much less important than the corresponding induced drag of an airplane wing. 

6.8. Effect of Compressibility 

Adequate treatment of the effect of compressibility would require very extensive 
research, and the remarks here made will be confined to a few fundamentals. 

large overall changes (not merely local changes) in the density of a fluid, 
flowing through a single row of blades would normally occur only in the case of turbines. 
Since the degree of reaction along the radius is practically always variable, tjie density 
at the exit of the stationary blade is also variable - often to such an extent that due 
account must be taken of the variation in making calculations. At the exit of the moving 
blade the differences of density are so small that they can always be neglected. 

The nature of the cross-flows induced by these effects may easily be deduced. 

Since the degree of reaction is smaller at the inner diameter than at the outer, there 
is a greater expansion on the inside, and a correspondingly lower density at the end of 
expansion. In the stationary blade grid, therefore, there is a certain divergence of 
the meridian Btreamlines near the inner diameter, and a certain convergence near the 
outer diameter. The cross-flow is therefore just opposite to that found in the numerical 


120 


2321 




example of, 6.7. Actually, one effect or the other will predominate, depending upon the 
variation of angle and upon the Mach Number. 

In general, It may be said that the effect of compressibility is always such as 
to make the degree of reaction along the blade more variable than it would be for an 
incompressible fluid and similar blades. The degree of reaction varies in this way be-» 
cause a certain radial pressure gradient is necessary In order to balance the centri¬ 
fugal force of the individual particles, or more correctly expressed. In order to impart 
to the fluid particles the normal acceleration required to enable them to remain on the 
same stream surface. 

Considering now the relations at the exit of the stationary blade grid of Figure 
29 , it is evident that the normal acceleration will be definitely less in this case than 
it would be for cylindrical stream surfaces, since in the meridian section the stream 
surface is "bent toward the other side" Just as in a section perpendicular to the axis 
of rotation. The degree of reaction in this case is. In fact, less variable than for 
cylindrical stream surfaces, i.e., than for the normal stage. Just the reverse would 
be the case if the meridian streamlines at the exit of the stationary blade were "convex 
toward the outside". 

The effect of compressibility is such that the meridian streamlines in the stationary 
blade grid are forced toward the outside. A path which is convex toward the inside for 
an incompressible fluid, such as is shown in Figure 29, becomes at least less noticeable, 
and under certain conditions may even have its curvature reversed. It has been previously 
shown, however, that in this case there can be dynamic equilibrium only if the degree 
of reaction along the radius varies considerably. This effect occurs for all blades, 
whether they are "correctly" shaped or not. 

In conclusion, the following observations are offered. 

(a) If the flow of a compressible fluid between two co-axial cylinders is with¬ 
out losses, and is produced solely by pressure forces from an original state of rest, 
the tangential velocity follows the law c u r = constant and the axial velocity the law 

c a z constant, Just as in the case of an incompressible fluid. This fact can easily be 

derived from the general laws of hydrodynamics, but it is also evident directly from 
consideration of the dynamic equilibrium of a fluid element. 

(b) The amount of fluid flowing through the annular cross-section of the meridian 
passage must theoretically be determined by an integration along the radius, taking due 
account of the variation of density. In the case of turbines, it is sufficiently ac¬ 
curate to assume that the density is constant over the entire annular cross-section, and 
is equal to the existing density at the mean diameter. Numerical examples show, how¬ 
ever, that such calculations made for stages of a type not well suited to treatment by 
the present theory may not be reliable. This is true particularly of low-pressure steam 
turbine stages operating in a high vacuum. 


121 


2321 


ILLUSTBATIONS 


Fig. 1. Meridiankanal mit Meridianstromlinienbild 

Fig. 1. Meridian Passage and Meridian Streamline 
Pattern. 



Stage Element 


Principal 
Stream Surface 


Stufenelement 


,122 


2321 




















































Turbine 


i Compressor 


Turbine 



A P 


i h 


ad , 


AP'> 



a 


Fig. 2. Entropiadiagramme der Zustandsanderung im Stufenelement 

Fie. 2 Change of State in a Stage Element Shovrn on the 
^thaloy-Entrooy Chart 


123 


2321 











































Turbine 

Turbine 



% 


Cut 
U, > 


%- 


Cur 
U t • 


9 , 


Cat 
U f • 




Compressor 

Kompressor 



V 
Fig. 3. 
Fig. 3 


. Cut _ Cut ~ _ Cat — _ Cmi 

— • T i “C7 • 9t -nj- * 9t -uj 

Geschwindigkeitsdreiecke, allgemeinster Fall 

Velocity Diagrams (General Case; 


124 


2321 






















Turbine 


Turbine x r -T,-T>o 



Compressor 

Kompressor K-T,-T t <o 



Fig. 4 Velocity Diagrams for the Normal Stage Element 


125 


2321 







































a 

< 

o 

n 

r 

\ 


r 

r 

n 

1 


<- 

\ 


i 

\ 

y 

i 

1 

*> 

\ 

K. 

r\ 


r\ 

1 


i 

*-> 

1 

*-N 


— 

<} 

1 

o 

d 

1 

»s> 

\ 

ft 

O' 


"N 

"N 

"N 

Ul 

* 


"\ 

*> 

\ 

1 

■n 

"\ 

"N 

t 

k 

k 

S 

M 

1 

's. 

H 

\ 


\ 

v> 

O' 


j 

1 


~N 

~s 

K 

A 

\ 

K 

■n 

J 

A 

1 

0 

N 

1 

1/1 

v 

A 

\ 

i 

y 

\ 


M 

\ 


1 

ft 

O' 


s 

Jl 

A 

\ 

N 

\s 

4 

j 

* 


sJi 

<> 

\ 


-> 

\ 

w 

y 

A 

A 

\ 

1 

A 

<* 

/n 

* 

l 

r 

n 

r 

J 

o 


1 


oj 

\ 


V 

h 

I 

1 

<w< 

V 

s 

t 

N 


M 

r 

1 

1 

it 

i 

r 

d 

% 

I 


U 

Sw> 

V 

H 

o 

k 

V 


V. 

Si 

1 

N 

N 

S 

V 

1 

N 

~N 

"N 

-> 

I 


k 

~N 

k 

1 


it 

«< 

* 

<n 

CM 


O 

*<* 

1 

CM 

1 


126 


2321 


Ubersicht iiber Normalstufenelemente 



































































































































































































































Fig. 7. Schaufelgitter des Zahlenbeispiels 

Fig. 7 Blade Grid Used in Numerical 
Example 


127 


Schaufelgitter 
Blade Grid 


y 

Fig. 6. 
Fig. 6 
































X 






u 

^ 5 > 




Fig. 8. Geschwindigkeitsplan des Schaufelgitters 
Fig. 8 Velocity Diagram of Blade Grid 



Fig. 9. Abgewickeltes Stufenelement 

Fig. 9 Developed Stage Element 
128 


2321 































f(x) 



Fig. 10 Velocity Distribution Function Used in Numerical Example 



Fig. 11. Geschwindigkeitsdreiecke des allgemeinen zylindrischen Stufenelements 
Fig. 11 Velocity Diagram of the General Cylindrical Stage Element 

129 


2321 




































Fig. 12. Graphische Bestimmung einander zugeordneter Werte von und 

Fig. 12 Graphical Determination of the Associated Values of 

and jSk 




Fig. 13. Zur Herleitung der Spaltverlustformel 

Fig. 13 Diagram Used in Derivation of Clearance 
Loss Formula 


130 


2321 
























Fig. 14. Zur'Herleitung der Spaltverlustformel 

Fig. 14- Diagram Used in Derivation of Clearance 
Loss Formula 


z 



Fig. 15. Extrapolation auf Spaltweite Null 
Fig. 15 Extrapolation to Zero Clearance 


131 


2321 















a b 

Fig. 16. Zur Herleitung der Randverlustformel 

Fig. 16 Diagram Used in Derivation of Wall Loss 
Formula 


132 


2321 
































Turbine Kom pressor 




]EKdi -(1 +j)Had^Ha± 

Fig. 17. is-Diagramm der Zustandsanderung in mehrstufiger Turbomaschine 

Fig, 17 Change of State in a Multistage Turbo- 
machine. Shown on the Enthalpy- 
Entropy Chart 


133 


2321 
























T 



H ad - ABF„- ABODE 

Fig. 18. Ts-Diagramm der Expansion in mehrstufiger Turbine 

Fig. 18 Expansion in a Multistage Turbine Shown on the 
Temperature-Entropy Chart 


134 


2321 













Zero Stage Group I Group II 



Fig. 19 Multistage Blading. Division into Groups of Stages 


135 


2321 
























































=-0.037 


Turbine 

K C i 



Fig. 20. Velocity Diagrams (at Principal Diameter) for the 
Illustrative Numerical Example 


/36 


2321 


























y 




Fig. 21. Single Stationary Blade Grid in an Infinitely Long Meridian 
Paesoge with Cylindrical Bounding Walls 


/37 


2321 










Fig. 22. 

Fig. 22 


Fig. 23. 
Fig. 23 





Koordinatensystem, Richtung des Geschwiiidigkeits- und des Wirbelvektors 
sowie des Gradienten des Totaldruckes in den Zwischenraumen 
zvvischen den Schaufelkriinzen 

Coordinate System, Direction of the Velocity and Rotation Vectors, 
and Direction of the Gradient of the Total Pressure in the 
Clearance Space between the Rows of Blades 



Gruppe vollig gleichartiger Stufen, von inkompressibler Flussigkeit durchstromt 
Flov; of an Incompressible Fluid Through a Group of Identical Stages 


23 8 


2321 
















































Fig. 24. Zum Grenzubergang auf unendlich kleine Schaufelteilung 

Fig. 24 Transition to the Limiting Case of an Infinitely Small 
Blade Pitch 


139 


2321 





































































Fig. 25. Leitrad mit schiefgestellten, gebogenen Sehaufeln 
Fig. 25 Stationary Grid with Curved, Inclined Blades 

</ 



Fig. 26. Homogene Stufengruppe mit schiefgestellten Leitschaufeln 
Fig. 26 Homogeneous Group of Stages with Inclined Stator Blades 


140 


2321 













































Fig. 27. Kompressorschaufelung mit schiefgestellten Leitschaufeln, Zahlenbeispiel 6. 5. 

Fig. 27 Compressor Blading with Inclined Stator Blades (Numerical 
Example 6.5) 



Fig. 28. Theorie der wirbelbehafteten Stromung, Koordinatensystem, Kontro-llebenen 
Fig. 28 Theory of Turbulent Flow, Coordinate System and Reference Planes 

141 


2321 




















































Fig. 29. Meridianstromlinienbild, Verlauf von C fl1 , C a9 , , a? fur Reaktionsturbine 

mit kongruenten unverdrehten Schaufeln 

Fig. 29 Meridian Streamline Pattern, Variation of C gl , C a2 >Ofi» and Cf 2 
for a Reaction Turbine with Identical Unwarped Blades 


142 


2321 


















































Br o.iioo 

3 - 0,0795 

El 



■Q* -0? G 0,2 g f 0,4 

Fig. 30. Fehlerintegral $ in Funktion von B A 
Fig. 30 Integral F as a Function of B 1 


143 


2321 

















Inner Diameter, 
or Hub 



At Outer Diameter, 
or Tip 



1 


Fig. 31. Geschwindigkeitsdreiecke in verschiedenen Stromflachen fur Reaktionsturbine 
mit kongruenten unverdrehten Schaufeln 


Fi8 ' 31 ?H^?^ y i D i i T agramS J a ^ D J fferent Radil for Reaction Turbine with 
Identical Unwarped Blades 


144 


2321 













7,4 

<// 

1,3 


i t 2 


1,1 


l 

0 0,1 OJ 0,3 0,4 0,5 0,6 0,7 

-- 6 > 

Fig. 32. Verlauf des Reaktionsgrades langs der Schaufel 
Fig. 32 Variation of Degree of Reaction along the Blade 



145 


2321 























Model 

Nodell 


Equalizing Chamber 

A usgleichbehalter 


Diffuser 


Blower 


Dif fusor Transition Ventilator 

- r Piece 7 



Fig, 33. Windkanal nut Modellgitter 
Fig. 33 Wind Tunnel with Grid Model 


Measuring Plane2 Measuring Plane 1 Profile MCA 6309 



Fig. 34. Durchgemessenes Schaufelgitter 
Fig. 34 Experimental Blade Grid 


146 


2321 
































Fig. 35. Resultate einer Gittermessung 
Fig* 35. Grid Test Results 




ro 


147 


2321 


in g Plane 0 Measuring Plane 1 Measuring Plane 







































































































APPENDIX 

VIL Anhang. 



2321 


Table 1 Relation between Grid Energy Loss 

Coefficient and Friction Coefficient 




































































































































+ CN 


II 







II 







II 

x 



« (M 


CN 


cn 


II 


o 

X 


IN 




II 




CN 


LU 

U- 

< 

f- 


x 1 

O'for'- 

cn c- cn h-co O"o n o co 

O O iD 0"t 05 CO 00 co r» 
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00 O'O'O0 O CN CO 00 CN o 
CNt'-CNt^COOOCOOOO'O' 
O Cq —_ — cq cq CO CO^ O^ -rr 

cn' cn' cn cn' cn' cn cn' cn' cn' cn' 

2,550 

2,837 

3,135 

3,453 

3,785 

4,133 

4.500 I 

>F 

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O' © r — i co oo 
o o o o^o^o^ 
o' o' o' o' o' o' 

O'O — — — — oooo© 
oinocot'- - moiCN© 

O'O'O'OO — — — CNCN 

o o. o. — — — — — -q 
o' o' o' o' o' o' o' o o o 

O' — 00©CNO'© — r^CN 
OO'Nr-©00CN©O'CF 
COCOCOO'O'O'©©©© 

o o' o' o' o o o' o o o' 

r- — © O' con 
© o cn © o cn 
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o' o' o' o' o' C 

>s 

1,00 

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CM CN CN CN CqcN cqcqcqcq 

©_-<M<riO'©©r-aoa' 
eo cqcqcqcocqcqcqcqcq 

© — CN CO O' £ 
Tj- O' O' O' o < 
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149 

1 ' 

2321 


Table 2 Functions of Y 
















































TAFEL 3. Werte der Stufenzahlfunktion 




N 

T-CNcoo-mor'-coo 

©*—CNcOTfmof'-ooO' 

© — CNCOTfinOC-QC©© 
CNCNCNCNCNCNCNCNCNCNCO 


1 

+ 0,50 

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in^oooinn'-ooo 
cq oq co o^in r- r-~ co^ cq 

cn" co" in" o' 00" o'" —~ co" o r 

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rj*^© oiNoo^cno^in^ 
o'OcT O'" 7—< CN rf" in'f- O' © 
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©ooomco-©ooo + co 
cn 00 , + ©^0 cq rq q 0^ q Tq 
cn"co" in"r~"00"©"'—"co tt"0 00" 

CO CO CO CO CO 7 f 't 't 'T N- 

o 

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Cq rr 0 q cq tq CN © 7- iq 

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CNn-(NOOCOOO + C 3 "t't 

o_ rf 01 co^ 00^ cn 7— q 0^ 
in 0" t^-" O'" o" cn co" in 0 00" 

— — — — CNCNCNCNCNCN 

mom - O'— c-cNr~cooo 
in © rqcqcqoqcqrq-qNq© 
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o^o^cnoacno^o^cn 
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vnoo — Tfr-'ocooo'CN 
O © CO O O' cn 0 © CN^o 
co rt" 0" r-" 00" 0" 7—" cn" Tf" vn" 

— — — — — ^1 M CN CM CN 

moo-"tr~ocoo©cNin 
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0" 00" ©" ©" cn" co" '+" 0" r~" 00" ©" 

CNCNCNCOCOCOCOCOCOCOtJ' 

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M 00 00 00 O' 

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iq 00 0 co 0 cq t— rq t-q 
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cn" rf' in" 0" co O' © t-" co" rr" 

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©t—-TJ 7 -GoincN©r^-o.— 

© cn m rq cq cq q oq 7--^rq 
in" 0" co" ©" ©" cn" co m" uo c-" 00" 

CNCNCNCNCOCO.COCOCOCOCO 

ts» CQ< 

03 ''r' 
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<V N 

be ii 

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n * 
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+ 0,20 | 

OOMIOCO'-'COO 
'vinot^ooo'to't 
■+ o^k o^cs in 

7-' Cn" co" o' 0" t'-"' CO O' 

inr-oo© 7 — cNcoinot'-* 
cq uq tq 01 cn q 0^ oo_ cq cq 
cn" co" 't" in" r-" 00" ©" ©' cn" co" 

— — — — — — — CNCNCN 

©© — cOTfint—co© — cn 
' e* tq©^—^cqiqrqcq»q'qO s 
7 t" m" 0" 00 O' cd — cn" rr' in' 0 

CN CN CN CN CN CO CO CO CO CO CO 

00 

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CN—0—' — CN CO CO 

O' 00 O 10 ^ CO CN - 

cqiqtqcq—^cqiqr'- 0 
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o©©oor—oin' 3 -cocN 
CN ttiO* © Ct^O. 00 ^ 
cn" co" rf" in" 0' oo ©" ©" -— cn" 

— —— — — — — CNCNCN 

— ©©©oon-oin-o'cocN 
© cqcqqiqcq—cqin rqcq 
■*}•" in' o' ocT 0^ —' cnT co" rt*" in" 

CN CN CN CN CN CN CO CO CO CO CO 

o 

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cqiq©^oo i O_ s --^cq_iq'q 
7 —" cn"co" rr" o' P-" 00" ©" o' 

ocooocoor^-'+T-oo 
oq cq eq cq in rq oq © cn co^ 

7—" co" in" 0" r~" 06 0" cn" 
7— — — — — — — CNCNCN 

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in tqco ©^cqcqiqtqcqcqcq 
co rr" in" r-" co" ©" © —" cn" in 

CN CN CN CN CN CN CO CO CO CO CO 

C 0 
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be © 
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rr 

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V—< 

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0 tq©_cqcN cqq q tqo^ 

7—" cn co" in 0" r- 00* 0" 0" 

— — — — — — — — cmcn 

cocor-cNr- — o — inoin 

©„ ^1 cq q q cq ©_ *-q cq -q iq 
co" rf in' 0 ocT © —' cn" CO rf' 

CN CN CN CN CN CN CN CO CO CO CO 

°.<o 

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vn t'0 cn m 0 cn 10 
cn cq iq o_ tq oq © q cq 

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n-ocomooocNinr-o 
cq 0 0^ tq oq 0 7- cn cq 0 

7—" cn" cf rr" m" i"-" 00 © 0" 7—~ 

7—7 7—7 7—7 7—7 7—7 —7 7—7 —-7 CN CN 

CNinn-ocNinr'-ocNinr— 
o^tqco © —qqco^iqNqrqGq 

cn" co"^ f"o"r- "00" O' CD —"CN m" 

CN CN CN CN CN CN CN CO CO CO CO 

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— 4 

:© £ 

II o 

T—• 

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— — — — CNCNCNCOrf' 
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—" cn" co" rr" in" 0" r—" 00" 0" 

'f’tmincN'O'O'Or't^ 

7-q cq cq-q iq Nq rq oo^ cq cq 
»—' cn" cb~ rf" o o"r-" co" © 7—" 

1 '-C 0 c 000 ©©©©© 0 © 

-1 ^ T- i r i. 

cn" co" -rf in' 0" r-" 00" a" — " cn" co 

CN CN CN CN CN CN CN CN CO CO CO 

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Ol 

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cq© © *qcqcq^q'q»n o_ 
0" 7—" co rf" in" o' r~“ 00" ©" 0" 

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Nl 

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CO CN O O rr 00 CN O O 
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m©cot --- 

rf •rf iqiqo^vqo rqtqoq 

0 " 7 -" cn" co" in" 0" t— " 00" ©" 

m©coc~ — in©cot-> — m 
oq cq ©^ © <© ©, © — — cq cq 
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N 

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©-cNco^inon-oo© 

© — CNcoTfin©t'-co©o 
CNCNCNCNCNCNCNCNCNCNCO 

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M 

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CNOOrrOOdOOrCO 
cqoooqoq_tqtq© 0^0^ 

0" —" of co~ rf iff 0’ c-" 00 

0 

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iqiq-q-q q co^ co^ cq cq cq 
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r^co©m — t^-co©m — r- 
7-^ — ©^ © cq© <qoo© cqtq 

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rf © © — co in r— © cn 
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0" —" cn" co" V" ino'n»'oo" 

0 

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cq ©^ cq oq tq q 0 iq^f 

O'" 0" 0" ci cn rt m" 0" r-" 

r-© — coinoo©CNrf'0© 

cq cn cq 7—0 ©^ <q oo_ iq ©_ q 

00 © ©" —" cn" cn" <rf rr" m" ©" n-" 

— — CNCNCNCNCNCNCNCNCN 

+ 

Nj 

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1 

0 co 0 0 cn m O'cm m 

— — — — CNCNCNCOCO 

oqtq©,iqrqcqcNqcq 
0 " t-*' cn" co rr" in" ©" r-" 00 

© CN 

cO'Tin-inoooh'h 
©_ °o rq vq iq cq cq r--cq 

00" O'" o' —" cm" n 7 t in" 0* c-" 

• 

n-oocooo©©©©©©© 
cq oq iq iq rq cq cq cq —^ © 

r-" 00 © ©" —"cn"co" rr" in"o' r-" 

— — — CNCNCNCNCNCNCNCN 

+ 

M 

|| 

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©" 

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c- in rr cn — O' 00 0 m 
tq© inrqcoq© © oq 
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in © 

CO CN O © 00 O Tf CO t—< 0 

tq 0^ in cq cq© v 0^ °q rq 
00 © © t—' cT co~ 7^' Tf" in' o' 

00 h- in rr cn 7-. ©oo©inco 
uq rq eq cn — © oq tq © in rq 

c-" 00" ©" 0 " —' cn" cn" co" rr" in" ©" 

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11 

M OQ 

rf 

CD 

1 

O + Oh-COO'OCOO' 
'TO^coor'COO'O 
CO^ CN^ 0 0^ oo_ 0^ 

0" 7 -" cn" co" m" in" m" 0" r—" 

O CN 

roon-coon-coot^n 
iq'qcqT—cqcqrqO ^cq^ 

co O' cb *— " cn" cn" co" -'t" in 0" 

ot—co©t'-co©t--co©r'- 
cqcqcqoqvqiqrrcq —^ ©^ 00^ 

r»" 00" co (© cT — cn" co rr" ©" ©" 

— — — — CNCNCNCNCNCNCN 



-0,16 

OMOOO'OincOMO 

0 in 0 't ot o>-i 1 0 
r- inrf i n)O, 0 '^,o + 

o' -* cm" cn ■q r 7 t in c r— " 

GO C~ 

C'OO-'-OOCOGOCOGOCOOO 

cq — cq oq rq q -q cq —^ ©^ 
00" 0^ co CD —" cn" co' 'ft' in' m" 

CNt — CNt'-CNt — CN© 7-7 © — 

oq© ©^qcN ©^cqtq© qco 
©" r-" 00" ©" © — " —" cn" co" rr" in" 

— — — — CNCNCNCNCNCNCN 



N 

7 . cn co ^ m 0 r'00 O' 

© — cn co rf n 0 n co 0 

© — cNmrr©©n-oo ©0 

CNCNCNCNCNCNCNCNCNCNCO 


150 


2321 


Table 3 Values of Function Used in Obtaining Number of Stages 





















































TAFEL 4. 


Table 4 Heat Recovery and Heating Loss Factor 



; IS 2 3 ♦ 5 7 to 1 IS 2 3 4 5 7 W 

Pit) __ Pm 

Piz\ Pin 



2321 


151 
















































































































































































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